Geography 625

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Geography 625
Intermediate Geographic Information Science
Week3 Fundamentals Maps as outcomes of process
Instructor Changshan Wu Department of
Geography The University of Wisconsin-Milwaukee Fa
ll 2006
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Outline

• Introduction
• Processes and the patterns
• Predicting the pattern generated by a process
• More definitions
• Stochastic processes in lines, areas, and fields
• Conclusion

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1. Introduction

• Maps as outcomes of process
• Maps have the ability to suggest patterns in the
  phenomena they represent.
• Patterns provide clues to a possible causal
  process.
• Maps can be understood as outcomes of processes.

Map
Processes
Patterns
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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
x
2
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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.
• This chance element seems inherent in processes
  involving the individual or collective results of
  human decisions.
• Some spatial patterns are the results of
  deterministic physical laws, but they appear as
  if they are the results of chance process.

z 2x 3y d
Where d is a randomly chosen value at each
location, -1 or 1.
y
2
2
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
Created Random numbers from Excel Int(10
Rand()) Use these numbers as x and y
coordinates Repeat this process
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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)

• 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.
• 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
Process (Complete spatial randomness)
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3. Predicting the Pattern Generated By a Process
Complete spatial randomness (CSR)

• Equal probability
• Independence

P (event A in Yellow quadrat) 1/8 P (event A
not in Yellow quadrat) 7/8
A
P (event A only in the Yellow quadrat) P (event
A in Yellow quadrat and other events not in the
Yellow quadrat)
B
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
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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
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(AB only)
P(AC only) P(IJ only) (no. possible
combinations of two events)
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
In our case, n 10, and k 2
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3. Predicting the Pattern Generated By a Process
Complete spatial randomness (CSR)
P (k events)
A
B
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
A
B
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.
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.