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

Week3: Fundamentals: Maps as outcomes of process



Outline

1. Introduction2. Processes and the patterns 3. Predicting the pattern generated by a

process4. More definitions5. Stochastic processes in lines, areas, and

fields6. Conclusion



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



2. Process and the Patterns

A spatial process is a description of how a spatial pattern might be generated.

Z = 2x + 3y Where x and y are two spatial coordinatesz is the numerical value for a variable

x

y

Deterministic: it always produce the same outcome at each location.

2

2




Z = 2x + 3y

Deterministic

x

y



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.

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2

x

y

Stochastic




y

Stochastic: two realizations of z= 2x + 3y ± 1


x x

y




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0 2 4 6 8 10

Created Random numbers from ExcelInt(10 * Rand())

Use these numbers as x and y coordinates

Repeat this process

Dot map with randomly distributed points



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.




AB

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)



AB



1) Equal probability2) Independence

P (event A in Yellow quadrat) = 1/8P (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)

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1

A B C D E F G H I J





AB

P (one event only)= P (event A only) + P (event B only) + … + P (event J only)= 10 × P (event A only)

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110





AB

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))

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A B C D E F G H I J





AB

P ( two events in Yellow quadrat) = P(A&B only) + P(A&C only) + … + P(I&J only)

=(no. possible combinations of two events) ×82

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1

How many possible combinations?





AB

The formula for number of possible combinations of k events from a set of n events is given by

k

n

knk

nC nk )!(!

!

1...)2()1(! nnnn

In our case, n = 10, and k = 2





AB

P (k events) =

kk

kk

k

kk

C

10

1010

8

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1

)!10(!

!10

8

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1

knk ppk

nknP

)1(),(

p = quadrat area / area of study region





AB

Binomial distributionknk

x

x

xk

nxnkP

11),,(

x is the number of quadrats usedn is the number of eventsk is the number of events in a quadrat





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.

!)(

k

ekP

k

x

n e is a constant, equal to 2.7182818



3. Predicting the Pattern Generated By a ProcessComplete spatial randomness (CSR)

Comparison between binomial and Poisson distribution



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.

Geography 625

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