geography 625

University of Wisconsin-Milwaukee

Geographic Information Science

Geography 625

Intermediate Geographic Information Science

Instructor: Changshan WuDepartment of GeographyThe University of Wisconsin-MilwaukeeFall 2006

Week 10_11: Area objects and spatial autocorrelation



Outline

1. Introduction2. Geometric properties of areas3. Spatial autocorrelation: joins count approach4. Spatial autocorrelation: Moran’s I5. Spatial autocorrelation: Geary’s C6. Spatial autocorrelation: weight matrices7. Local indicators of spatial association (LISA)8. Spatial regression9. Spatial expansion method10. Geographical weighted regression



1. Introduction

1. Natural areas: self-defining, their boundaries are defined by the phenomenon itself (e.g. lake, land use)

- Types of area object

Lake map



1. Introduction- Types of area object

2. Imposed areas: area objects are imposed by human beings, such as countries, states, counties etc. Boundaries are defined independently of any phenomenon, and attribute values are enumerated by surveys or censuses.

Potential Problems

1. may bear little relationship to underlying patterns2. Arbitrary and modifiable3. Danger of ecological fallacies (aggregated format)




3. Raster: space is divided into small regular grid cells.

In a raster, the area objects are identical and together cover the region of interest.

Each cell can be considered an area object. Raster data are always used to represent continuous phenomenon.

Squares

Hexagons




Planar enforced: area objects mesh together neatly and exhaust the study region, so that there is no holes, and every location is inside just a single area.Not planar enforced: the areas do not fill or exhaust the space, the entities are isolated from one another, or perhaps overlapped



2. Geometric Properties of Areas- Area

x

y

(x1, y1)

(x2, y2)

(x3, y3)(x4, y4)

n

iiiii yyxxArea

1112

1 ))((

Assume x1 = xn+1



2. Geometric Properties of Areas- Skeleton

The skeleton of a polygon is a network of lines inside a polygon constructed so that each point on the network is equidistant from the nearest two edges in the polygon boundary.



2. Geometric Properties of Areas- Skeleton

center

n

iixx

1

ˆ

n

iiyy

1

ˆ

Arithmetic center

Center derived by skeleton analysis



2. Geometric Properties of Areas- Shape

Shape: a set of relationships of relative position between points on their perimeters.

In ecology, the shapes of patches of a specified habitat are thought to have significant effects on what happens and around them.

In urban studies, urban shapes change from traditional monocentric to polycentric sprawl




Perimeter: P

Area: a

Longest axis: L1

Second axis: L2

The radius of the largest internal circle: R1

The radius of the smallest enclosing circle: R2



Compactness ratio


2/ aa

a is the area of the polygona2 is the area of the circle having the same perimeter (P) as the object

What is the compactness ratio for a circle?What is the compactness ratio for a square?




Other measurements

Elongation ratio: L1/L2

Form ratio: 21/ La



3. Spatial autocorrelation- Joins count approach

Developed by Cliff and Ord (1973) in their book: Spatial Autocorrelation

The joins count statistic is applied to a map of areal units where each unit is classified as either black (B) or white (W).

The joins count is determined by counting the number of occurrences in the map of each of the possible joins (e.g. BB, WW, BW) between neighboring areal units.




Possible joins:

JBB: the number of joins of BBJWW: the number of joins of WWJBW: the number of joins of BW or WB

Neighbor definition

Rook’s case: four neighbors (North-South-West-East)Queen’s case: eight neighbors (including diagonal neighbors)




Statistical tests for spatial correlation

Independent Random Process (IRP)

wBBW

WWW

BBB

pkpJE

kpJE

kpJE

2)(

)(

)(2

2

Where k is the total number of joins on the mappB is the probability of an area being coded BpW is the probability of an area being coded W

Mean





The expected standard deviations are as follows

22

432

432

)2(4)(2)(

)2(3)(

)2(2)(

WBWBBW

WWWWW

BBBBB

ppmkppmksE

pmkmpkpsE

pmkmpkpsE




n

iii kkm

1

)1(2

1

ki is the number of joins to the ith area

m = 0.5 [(4×2×1) + (16×3×2)+(16×4×3)]

corners edges center

= 148





)(

)(

)(

)(

)(

)(

WW

WWWWWW

BW

BWBWBW

BB

BBBBBB

sE

JEJZ

sE

JEJZ

sE

JEJZ

A large negative Z-score on

JBW indicates positive autocorrelation since it indicates that there are fewer BW joins than expected.

A large positive Z-score on JBW is indicative of negative autocorrelation.



3. Spatial autocorrelation- Joins count approach (example)

B: BushW: Gore

State-level results for the 2000 U.S. presidential election




Adjacency (joins) matrix: if two states share a common boundary, they are adjacent.




Bush48,021,500pB = 0.49885

Gore48,242,921pw = 0.50115



4. Spatial autocorrelation- Moran’s I

Limitations of Joins Count Statistics

1. It can only be applied on binary data

2. Although the approach provides an indication of the strength of autocorrelation present in terms of Z-scores, it is not readily interpreted, particularly if the results of different tests appear contradictory

3. The equations for the expected values of counts are fairly formidable.




n

i

n

jij

n

i

n

jjiij

n

ii w

yyyyw

yy

nI

1 1

1 1

1

2

))((

)(

wij =1 If zone i an zone j are adjacent

0 otherwise




2 0

2 0

n

i

n

jij

n

i

n

jjiij

n

ii w

yyyyw

yy

nI

1 1

1 1

1

2

))((

)(

1 2

3 4

A =

0110

1001

1001

01101

2

3

4

1 2 3 4




2 0

0 2

1 2

3 4

n

i

n

jij

n

i

n

jjiij

n

ii w

yyyyw

yy

nI

1 1

1 1

1

2

))((

)(

A =

0110

1001

1001

01101

2

3

4

1 2 3 4




For Moran’s I, a positive value indicates a positive autocorrelation, and a negative value indicates a negative autocorrelation.

Moran’s I is not strictly in the range of -1 to +1.



5. Spatial autocorrelation- Geary’s C

n

i

n

jij

n

i

n

jjiij

n

ii w

yyw

yy

nC

1 1

1 1

2

1

2 2

)(

)(

1


0 otherwise

Proposed by Geary’s contiguity ratio C




n

i

n

jij

n

i

n

jjiij

n

ii w

yyw

yy

nC

1 1

1 1

2

1

2 2

)(

)(

12 0

0 2

1 2

3 4

A =

0110

1001

1001

01101

2

3

4

1 2 3 4




The value generally varies between 0 and 2.

The theoretical value of C is 1 under independent random process. values less than 1 indicate positive spatial autocorrelation while values greater than 1 indicate negative autocorrelation.



6. Spatial autocorrelation- Other Weighting Matrices

1. Using distance

0

zij

ij

dw

Where dij < D and z < 0

Where dij > D




2. Using the length of shared boundary

i

ijij l

lw

Where li is the length of the boundary of zone ilij is the length of boundary shared by area i and j




3. Using both distance and the length of shared boundary

i

ijzij

ij l

ldw



7. Spatial autocorrelation- Local Indicators

Global statistics tell us whether or not an overall configuration is autocorrelated, but not where the unusual interactions are.

Local indicators of spatial association (LISA) were proposed in Getis and Ord (1992) and Anselin (1995).

These are disaggregate measures of autocorrelation that describe the extent to which particular areal units are similar to, or different from, their neighbors.




Local Gi

n

i i

ij iij

iy

ywG

1

Used to detect possible nonstationarity in data, when clusters of similar values are found in specific subregions of the area studied.



2 0

0 2

1 2

3 4

n

i i

ij iij

iy

ywG

1



0 otherwise




Local Moran’s I

ij

jijii zwzI

syyz ii /)( Where

W matrix is a row-standardized (i.e. scaled so that each row sums to 1)




ij

jijii zwzI

syyz ii /)(

=

02/12/10

2/1002/1

2/1002/1

02/12/10

2 0

0 2

1 2

3 4

0110

1001

1001

01101

2

3

4

1 2 3 4




Local Geary’s C

2)( jiiji yywC

2 0

0 2

1 2

3 4



8. Spatial regression Models

UXY Where U is a zero-mean vector of errors with variance-covariance matrix C

CUUE

UET

)(

0)(

If C = I, this is the ordinary least square (OLS) model



8. Spatial regression Models- Simultaneous autocorrelation model (SAR)

WUU

UXYIE

ET 2)(

0)(

WXWYX

XYWX

WUXY

)(

and

12 ))1()1((

)(

WW

UUECT

T



8. Spatial regression Models- Simultaneous autocorrelation model (SAR)

Software for SAR model

ArcView 3.2 + S-PlusR programming language



9. Spatial Expansion Models

Proposed by Casetti (1972)

...

...)(

)(...)()(

...

221101

22110

22110

zczccZb

xZbxZbxZbbY

xbxbxbbY

nn

nn

OLS model

Expansionmodel



10. Geographic Weighted Regression

...

...

...

22110

22110

ininiiiiiii

nn

xbxbxbbY

xbxbxbbY

OLS model

GWR model

The coefficients b vary with respect to the location i



10. Geographic Weighted Regression

Software

R programGWR package (free from Fotheringham)

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