Transcript No Slide Title

Lecture 16: Spatial Autocorrelation III
Topics:
2.3 Measures based on absolute adjacency
2.3.1 Area features
2.3.1.1 Geary Index (ratio/interval)
2.3.1.2 Moran Coefficient (ratio/interval)
2.3.1.3 Joint Count Statistics (categorical)
2.3.2 Other type of features
References:
Goodchild, Michael F., 1986. Spatial Autocorrelation,
CATMOG 47, Geo Books, Norwich, UK, 56 pp.
Griffith, Daniel A., 1987. Spatial Autocorrelation: A Primer,
Resource Publications in Geography, AAG, Washington, 82 pp.
Odland, John, 1987. Spatial Autocorrelation,
SAGE Pulications, Inc., Beverly Hills, CA, 85 pp.
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Outlines
2.3.1 Area Features
2.3.1.3 Joint Count Statistics:
Two categories (yes or no): black and white
(The Buffalo Figure)
1) Calculation:
Counting for number of WW, BB, BW joints
J = JWW+JBB+JBW
J is the total number of connections
2) Interpretation:
a) Under the random arrangement assumption:
(1) What is random arrangement
The population of spatial patterns for a given number
of events over a given number of area units is created
by rearranging the number of events over the area units
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2) Interpretation: (continued…)
a) Under the random arrangement assumption: (continued…)
(2) The expected numbers of different joints
Expected EWW = J NW(NW-1)/N(N-1)
Expected EBB = J NB(NB-1)/N(N-1)
Expected EBW = 2 J NB NW/N(N-1)
NW, NB are the occurrence of W and B events, respectively
(3) Testing:
Z BW 
For BW joints:
N
 BW  E BW 
[ L ( L  1)]N
i 1
i
i
N ( N  1)
J BW  E BW
 BW
N
B
NW

4[ J ( J  1)   Li ( Li  1)]N B ( N B  1) NW ( NW  1)
i 1
N ( N  1)(N  2)(N  3)
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2
 E BW
2) Interpretation: (continued…)
b) Under the random sampling assumption:
(1) What is a random sampling:
The population of spatial patterns for a given number of
events over a given number of area units is created by
sampling with replacement.
(2) The expected numbers of different joints
Expected EWW = J (NW)2/ N2
Expected EBB = J (NB)2/N2
Expected EBW = 2 J NB NW/N2
NW, NB are the occurrence of W and B events, respectively
(3) Testing
J BW  E BW
Z BW 
For BW joints:
 BW
 BW
N
N
N B NW
N B NW 2
 [2 J   Li ( Li  1)]
 4[ J   Li ( Li  1)][
]
2
2
N
N
i 1
i 1
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3) Example:
(1) The Pattern (The Figure)
(2) The Observed Joints (BW):
JBW = 6
(3) Interpretation:
(a) Under Random Arrangement:
EBW = (2x19x5x6)/11x10=10.3636
∑Li(Li-1) = (Table7.7)=114
σBW = 1.7721
ZBW = (6-10.3635)/1.7721= -2.462
at 95% significant level, the critical value of a
negative standard normal deivate is -1.645, thus
the BW joints is very unlikely to have occurred
by chance.
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3) Example: (continued …)
(3) Interpretation: (continued…)
(b) Under Random sampling:
EBW = (2x19x5x6)/11x11=9.42
∑Li(Li-1) = 114
σBW = 2.2323
ZBW = (6-9.42)/2.2323= -1.532
at 95% significant level, the critical value of a
negative standard normal deivate is -1.645, thus
the BW joints is likely to have occurred
by chance.
2.3.2 Other Features
2.3.2.1 Point features (Points to Areas Conversion Figure)
2.3.2.2 Linear features
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Questions
1. How is similarity in attribute defined in computing joint count
statistics and how is similarity in location defined?
2. What is the main difference between Joint Counts and Moran
Coefficient? When would you apply the Joint Counts statistics?
3. What is the random arrangement assumption and when should
you apply it?
4. What is the random sampling assumption and when should you
apply it? What is the difference between the two assumptions?
5. How would you measure spatial autocorrelation using the absolute
adjacency approach for point or linear features?
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