Point Pattern AnalysisPoint Pattern AnalysisChapter 4Chapter 4

Geographic Information AnalysisGeographic Information AnalysisBy David O’ Sullivan and David J. UnwinBy David O’ Sullivan and David J. Unwin

Introduction to Point Pattern Introduction to Point Pattern AnalysisAnalysis

Simplest Possible Spatial DataSimplest Possible Spatial Data-A point pattern is a set of events in a study -A point pattern is a set of events in a study

regionregion

-Each event is symbolized by a point object-Each event is symbolized by a point object

-Data are the locations of a set of point objects-Data are the locations of a set of point objects

ApplicationsApplications-Hot-spot analysis (crime, disease)-Hot-spot analysis (crime, disease)

-Vegetation, archaeological studies-Vegetation, archaeological studies

Introduction to Point Pattern Introduction to Point Pattern AnalysisAnalysis

Requirements for a set of events to Requirements for a set of events to constitute a point patternconstitute a point pattern

-Pattern should be mapped on a plane-Pattern should be mapped on a plane

-Study area determined objectively-Study area determined objectively

-Pattern is a census of the entities of interest-Pattern is a census of the entities of interest

-One-to-one correspondence between objects and-One-to-one correspondence between objects and

eventsevents

-Event locations are proper-Event locations are proper

Introduction to Point Pattern Introduction to Point Pattern AnalysisAnalysis

Point DensityPoint Density-First-order effect: Variation-First-order effect: Variation of intensity of a processof intensity of a process across spaceacross space-Number of events per unit-Number of events per unit areaarea-Absolute location-Absolute location

Point SeparationPoint Separation-Second-order effect:-Second-order effect: Interaction betweenInteraction between locations based on distancelocations based on distance between thembetween them-Relative location-Relative location

• Describing a point pattern

Introduction to Point Pattern Introduction to Point Pattern AnalysisAnalysis

Descriptive statistics to provide Descriptive statistics to provide summary descriptions of point summary descriptions of point patternspatterns

-Mean center-Mean center

-Standard Distance-Standard Distance

Density-Based Point Pattern Density-Based Point Pattern MeasuresMeasures

First-order effectFirst-order effect Sensitive to the definition of the Sensitive to the definition of the

study areastudy area

Density-Based Point Pattern Density-Based Point Pattern MeasuresMeasures

Quadrant count methodsQuadrant count methods-Record number of events of a pattern in a set of-Record number of events of a pattern in a set of

cells of a fixed sizecells of a fixed size

-Census vs. Random-Census vs. Random

Density-Based Point Pattern Density-Based Point Pattern MeasuresMeasures

Kernel-density estimationKernel-density estimation-Pattern has a density at any location in the study -Pattern has a density at any location in the study

regionregion

-Good for hot-spot analysis, checking first-order-Good for hot-spot analysis, checking first-order

stationary process, and linking point objects tostationary process, and linking point objects to

other geographic dataother geographic data

Naive methodNaive method

Distance-Based Point Pattern Distance-Based Point Pattern MeasuresMeasures

Second-order effectSecond-order effect Nearest-neighbor distanceNearest-neighbor distance

-The distance from an event to the nearest event in-The distance from an event to the nearest event in

the point patternthe point pattern

Mean nearest-neighbor distanceMean nearest-neighbor distance-Summarizes all the nearest-neighbor distances by -Summarizes all the nearest-neighbor distances by

aa

single mean valuesingle mean value

-Throws away much of the information about the-Throws away much of the information about the

patternpattern

Distance-Based Point Pattern Distance-Based Point Pattern MeasuresMeasures

G functionG function-Simplest-Simplest

-Examines the cumulative frequency distribution -Examines the cumulative frequency distribution ofof

the nearest-neighbor distancesthe nearest-neighbor distances

-The value of G for any distance tells you what-The value of G for any distance tells you what

fraction of all the nearest-neighbor distances in thefraction of all the nearest-neighbor distances in the

pattern are less than that distancepattern are less than that distance

Distance-Based Point Pattern Distance-Based Point Pattern MeasuresMeasures

F functionF function-Point locations are selected at random in the -Point locations are selected at random in the

studystudy

region and minimum distance from point location toregion and minimum distance from point location to

event is determinedevent is determined

-The F function is the cumulative frequency-The F function is the cumulative frequency

distribution distribution

-Advantage over G function: Increased sample -Advantage over G function: Increased sample sizesize

for smoother curvefor smoother curve

Distance-Based Point Pattern Distance-Based Point Pattern MeasuresMeasures

K functionK function-Based on all distances between events-Based on all distances between events

-Provides the most information about the pattern-Provides the most information about the pattern

Distance-Based Point Pattern Distance-Based Point Pattern MeasuresMeasures

Problem with all distance functions Problem with all distance functions are edge effectsare edge effects

Solution is to implement a guard Solution is to implement a guard zonezone

Assessing Point Patterns Assessing Point Patterns StatisticallyStatistically

Null hypothesisNull hypothesis-A particular spatial process produced the observed-A particular spatial process produced the observed

pattern (IRP/CSR)pattern (IRP/CSR)

SampleSample-A set of spatial data from the set of all possible-A set of spatial data from the set of all possible

realizations of the hypothesized processrealizations of the hypothesized process

TestingTesting-Using a test to illustrate how probable an observed-Using a test to illustrate how probable an observed

value of a pattern is relative to the distribution of valuesvalue of a pattern is relative to the distribution of values

in a sampling distributionin a sampling distribution

Assessing Point Patterns Assessing Point Patterns StatisticallyStatistically

Assessing Point Patterns Assessing Point Patterns StatisticallyStatistically

Quadrant countsQuadrant counts-Probability distribution for a quadrant count-Probability distribution for a quadrant count

description of a point pattern is given by a Poissondescription of a point pattern is given by a Poisson

distributiondistribution

-Null hypothesis: (IRP/CSR)-Null hypothesis: (IRP/CSR)

-Test statistic: Intensity (-Test statistic: Intensity (λλ))

-Tests: Variance/mean ratio, Chi-square-Tests: Variance/mean ratio, Chi-square

Nearest-neighbor distancesNearest-neighbor distances-R statistic-R statistic

Assessing Point Patterns Assessing Point Patterns StatisticallyStatistically

G and F functionsG and F functions-Plot observed pattern and IRP/CSR pattern-Plot observed pattern and IRP/CSR pattern

Assessing Point Patterns Assessing Point Patterns StatisticallyStatistically

K functionK function-Difficult to see small differences between -Difficult to see small differences between

expectedexpected

and observed patterns when plottedand observed patterns when plotted

-Develop another function L(d) that should equal-Develop another function L(d) that should equal

zero if K(d) is IRP/CSRzero if K(d) is IRP/CSR

-Use computer simulations to generate IRP/CSR-Use computer simulations to generate IRP/CSR

(Monte Carlo procedure)(Monte Carlo procedure)

Critiques of Spatial Statistical Critiques of Spatial Statistical AnalysisAnalysis

Peter GouldPeter Gould-Geographical data sets are not samples-Geographical data sets are not samples

-Geographical data are not random-Geographical data are not random-Geographical data are not independent random-Geographical data are not independent random-n is always large so results are almost always-n is always large so results are almost always

statistically significantstatistically significant-A null hypothesis of IRP/CSR being rejected -A null hypothesis of IRP/CSR being rejected

meansmeans any other process is the alternative hypothesisany other process is the alternative hypothesis

David HarveyDavid Harvey-Altering parameter estimates by changing study-Altering parameter estimates by changing study

region size often can alter conclusionsregion size often can alter conclusions