Module organization 

Introduction to Spatial Analysis

  

Basic concepts Analytic perspectives Space as container Space as indicator Spatial dependence/autocorrelation Global Moran’s I  Local Moran’s I 

II. Spatial analysis of lattice data



Stuart Sweeney



GEOG 172, Fall 2007

Spatial econometric models 2

Spatial analytic traditions: Spatial data types

Basic concepts

Basic concepts:

• Spatial continuous (fields)  Geostatistics





• Points (objects)  Point Pattern Analysis

Spatial process underlying observed spatial pattern Spatial variation can be decomposed into: 





• Irregular / Regular lattice (objects)  Spatial Econometrics

Space as container; Space as indicator 



Container: Location is label; Independent observations Indicator: Relative location is meaningful 

• Volume of interaction among areas  Spatial Interaction Modeling

4

Analytical perspectives on space

Analytical perspectives on space

Basic concepts: Utility of Geoda:



Discover explicable patterns in the data  Dynamic and interactive exploration  Descriptive measures and visual assessment  Assess local and global autocorrelation 

 Local 

Connectivity is used to define measures

3

Basic concepts



Large scale variation: mean of spatial process zs=f(Xs’B+es) Small scale variation: covariance of spatial process Cov(ei ,ej )=0 /

Space as container 

Assumes location is simply a label



Observations are independent

 Used



“LISAs”: spatial clusters

Space as indicator 

Spatial econometric models



 Fit

covariates and/or spatial lag to describe mean residuals and specify error dependence structure for inference.

to define region / sub-area

Relative location is meaningful Connectivity is used directly to define measures

 Assess

5

6

1

Analytical perspectives on space

Analytical perspectives on space: space as container

Scenario: House Price=f(sqft) 

Space as container

Is the increase in price per unit square foot the same in different regions of the city? Indicator variable used to define region.  Define metric or view based on region ID. 

7

Analytical perspectives on space

Spatial dependence/autocorrelation

Analytical perspectives on space 



Assumes location is simply a label  Used





Spatial dependence

Space as container 

8



to define region / sub-area

“Everything is related to…near things more related…” (Tobler 1970) Empirical outcome: spatial autocorrelation 

Observations are independent



Space as indicator



Relative location is meaningful  Connectivity is used directly to define measures





1. Process variation spatially varying mean 2. Spatial covariance yi=f(yj), i=1,..,n and i=j

Measures  

Many alternatives; spatial data types Areal data: Moran’s-I

9

Spatial dependence/autocorrelation

10

Spatial dependence/autocorrelation

Aside:

Spatial Autocorrelation

• Spatial dependence: yi=f(yj), i=1,..,n and i=j



- frame dependence (MAUP)



- spatial process

 

• Spatial heterogeneity: yi = xi Bi + ei



- specification issue

yi=f(yj), i=1,..,n and i=j What is form of f(*)? Tobler’s law: f(*) function of proximity Operationalize as connectivity W, weight matrix:

- spatial regimes (areas with similar functional relationships)

11

12

2

Spatial dependence/autocorrelation

Spatial dependence/autocorrelation

Recall:

1

1

2

3 5

Binary W matrix:

6

7

4

2

Row standardized W matrix:

5

3 7

Spatial Autocorrelation (cont.)

4

Connectivity example: Map of seven areas

6

Share border: Neighbor(1)={2,3,4,7} Neighbor(2)={1,3} .. Neighbor(7)={1,3,6}

~ W=

0 1 1 1 0 0 1

1 0 1 0 0 0 0

1 1 0 1 1 1 1

1 0 1 0 1 0 0

0 0 1 1 0 1 0

0 0 1 0 1 0 1

1 0 1 0 0 1 0 13

Spatial dependence/autocorrelation

14

Spatial dependence/autocorrelation

Geoda: W matrix

Demo I: Creating W matrix   



Read shapefile directly Can view properties of W matrix Can easily create multiple for sensitivity analysis Can open and directly edit .GAL file

15

16

Spatial dependence/autocorrelation

Activity: Map to W

Application: Working with W 



Select: Tools>Weights>Create Input file: C:\temp\sbreal_tract_p.shp Output file: C:\temp\sbtrct_rook.gal  Select “rook”, click on create



 

 

 

Select: Tools>Weights>Properties Open sbtrct_rook.gal in a text editor (Notepad)



17

Draw an outline map containing eight areas/regions. Write the numbers 1 through 8 in the eight areas. Give the map to your neighbor. Write down the first two rows of the W matrix using your neighbors map. Use a rook contiguity rule. Exchange maps with a different neighbor and check each others work. 18

3

Spatial dependence/autocorrelation

Spatial dependence/autocorrelation

Spatial Autocorrelation 

W, weight matrix:



Suppose,

y1

y2

y7

y4

y5

y3

Spatial Autocorrelation (cont.) Binary W matrix:

Row standardized W matrix:

y=

y6 19

Spatial dependence/autocorrelation

20

Spatial dependence/autocorrelation

Spatial Autocorrelation (cont.)

Spatial Autocorrelation (cont.) Plot Wy, y: Wy

y Slope indicates degree of association between y values and average neighboring values 21

Global Moran’s

Global Moran’s

Moran’s-I and scatterplot Convert raw scores, y, to standard scores, z 



 

Z=(y-y)/sd(y)

I=z’Wz / z’z 



Moran’s-I and scatterplot

Slope of line fit to scatter of Wz, z

I>0, positive spatial autocorrelation I0, positive spatial autocorrelation I0 mean?  Inference: Is significantly different from 0? Ho: I=0, Ha:I>0 

Observed Moran’s I = 0.2486

I{2} =0.0232

I{3} =0.0525

29

30

5

Global Moran’s

Global Moran’s

Geoda: Moran’s-I, interpretation

Observed Moran’s I = 0.2486

31

Local Moran’s

Local Moran’s

LISA and Local Moran’s-I 





LISA concept

I=z’Wz / z’z 

32



Slope of line fit to scatter of Wz, z Global measure: I>0, I