Transcript Document

Geographic Information Science
Geography 625
Intermediate
Geographic Information Science
Week3: Fundamentals: Maps as outcomes of process
Instructor: Changshan Wu
Department of Geography
The University of Wisconsin-Milwaukee
Fall 2006
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Outline
1. Introduction
2. Processes and the patterns
3. Predicting the pattern generated by a
process
4. More definitions
5. Stochastic processes in lines, areas, and
fields
6. Conclusion
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1. Introduction
Maps as outcomes of process
1. Maps have the ability to suggest patterns in the
phenomena they represent.
2. Patterns provide clues to a possible causal process.
3. Maps can be understood as outcomes of processes.
Processes
Patterns
Map
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2. Process and the Patterns
A spatial process is a description of how a spatial pattern
might be generated.
Deterministic: it always produce the same outcome at each location.
Z = 2x + 3y
Where x and y are two spatial coordinates
z is the numerical value for a variable
y
2
2
x
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2. Process and the Patterns
y
Deterministic
Z = 2x + 3y
x
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2. Process and the Patterns
Stochastic
 More often, geographic data appear to
be the result of a chance process, whose
outcome is subject to variation that
cannot be given precisely by a
mathematical function.
z= 2x + 3y + d

This chance element seems inherent in
processes involving the individual or
collective results of human decisions. y

Some spatial patterns are the results of
deterministic physical laws, but they
appear as if they are the results of
chance process.
Where d is a randomly
chosen value at each
location, -1 or +1.
2
2
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x
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2. Process and the Patterns
Stochastic: two realizations of z= 2x + 3y ± 1
y
y
x
x
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2. Process and the Patterns
Dot map with
randomly
distributed points
10
9
8
7
Created Random
numbers from Excel
Int(10 * Rand())
6
5
4
3
Use these numbers as x
and y coordinates
Repeat this process
2
1
0
0
2
4
6
8
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3. Predicting the Pattern Generated By a Process
What would be the outcome if there were absolutely no
geography to a process (completely random)?
Independent random process (IRP)
Complete spatial randomness (CSR)
1.
Equal probability: any point has equal probability of being in any
position or, equivalently, each small sub-area of the map has an equal
chance of receiving a point.
2.
Independence: the positioning of any point is independent of the
positioning of any other point.
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3. Predicting the Pattern Generated By a Process
Complete spatial randomness (CSR)
Event: a point in the map, representing an incident.
Quadrats: a set of equal-sized and nonoverlapping areas
Pattern
A
B
Process
(Complete spatial randomness)
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3. Predicting the Pattern Generated By a Process
Complete spatial randomness (CSR)
1) Equal probability
2) Independence
A
B
P (event A in Yellow quadrat) = 1/8
P (event A not in Yellow quadrat) = 7/8
P (event A only in the Yellow quadrat)
= P (event A in Yellow quadrat and other
events not in the Yellow quadrat)
1 7 7 7 7 7 7 7 7 7
         
8 8 8 8 8 8 8 8 8 8
A B C D E F G H I J
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3. Predicting the Pattern Generated By a Process
Complete spatial randomness (CSR)
P (one event only)
= P (event A only) + P (event B only) + … +
P (event J only)
= 10 × P (event A only)
A
B
1 7 7 7 7 7 7 7 7 7
 10          
8 8 8 8 8 8 8 8 8 8
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3. Predicting the Pattern Generated By a Process
Complete spatial randomness (CSR)
P (event A & B in Yellow quadrat) = 1/8 ×1/8
P (event A & B in Yellow quadrat only) =
P ((event A & B in Yellow quadrat) and (other events
not in Yellow quadrat))
A
B
1 1 7 7 7 7 7 7 7 7
         
8 8 8 8 8 8 8 8 8 8
A B C D
E F
G H
I J
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3. Predicting the Pattern Generated By a Process
Complete spatial randomness (CSR)
P ( two events in Yellow quadrat)
= P(A&B only) + P(A&C only) + … + P(I&J only)
2
1  7
=(no. possible combinations of two events) ×    
8  8
A
B
How many possible combinations?
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3. Predicting the Pattern Generated By a Process
Complete spatial randomness (CSR)
The formula for number of possible
combinations of k events from a set of n
events is given by
A
B
 n
n!
C 
  
k!(n  k )!  k 
n
k
n! n  (n  1)  (n  2)  ...1
In our case, n = 10, and k = 2
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3. Predicting the Pattern Generated By a Process
Complete spatial randomness (CSR)
k
10  k
7
10  1 
P (k events) = Ck      
8 8
k
10  k
10!
1 7

   
k!(10  k )!  8   8 
A
B
 n k
P(n, k )    p (1  p) nk
k 
p = quadrat area / area of study region
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3. Predicting the Pattern Generated By a Process
Complete spatial randomness (CSR)
Binomial distribution
 n  1   x  1 
P(k , n, x)     

 k  x   x 
k
A
B
nk
x is the number of quadrats used
n is the number of events
k is the number of events in a quadrat
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3. Predicting the Pattern Generated By a Process
Complete spatial randomness (CSR)
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3. Predicting the Pattern Generated By a Process
Complete spatial randomness (CSR)
The binomial expression derived above is often not very
practical for serious work because of computation burden, the
Poisson distribution is a good approximation to the binomial
distribution.
P(k ) 
n

x
k e  
k!
e is a constant, equal to 2.7182818
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3. Predicting the Pattern Generated By a Process
Complete spatial randomness (CSR)
Comparison between
binomial and Poisson
distribution
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4. More Definitions
 The independent random process is mathematically
elegant and forms a useful starting point for spatial
analysis, but its use is often exceedingly naive and
unrealistic.
 If real-world spatial patterns were indeed generated
by unconstrained randomness, geography would have
little meaning or interest, and most GIS operations
would be pointless.
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