gist 4302: spatial analysis and modeling - point …...spatial point patterns further issues:...

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GIST 4302: Spatial Analysis and Modeling Point Pattern Analysis Guofeng Cao www.myweb.ttu.edu/gucao Department of Geosciences Texas Tech University [email protected] Fall 2014

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Page 1: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

GIST 4302: Spatial Analysis and ModelingPoint Pattern Analysis

Guofeng Cao

www.myweb.ttu.edu/gucao

Department of Geosciences

Texas Tech University

[email protected]

Fall 2014

Page 2: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Spatial Point Patterns

Characteristics:

� set of n point locations with recorded �events�, e.g., locations oftrees, disease or crime incidents S = {s1, . . . , s i , . . . , sn}

� point locations correspond to all possible events or to subsets ofthem

� attribute values also possible at same locations, e.g., tree diameter,magnitude of earthquakes (marked point pattern)W = {w1, . . . ,wi , . . . ,wn}

Analysis objectives:

� detect spatial clustering or repulsion, as opposed to completerandomness, of event locations (in space and time)

� if clustering detected, investigate possible relations with nearby�sources�

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 2/1

Page 3: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Spatial Point Patterns

Further issues:

� analysis of point patterns over large areas should take into accountdistance distortions due to map projections

� boundaries of study area should not be arbitrary

� analysis of sampled point patterns can be misleading

� one-to-one correspondence between objects in study area and eventsin pattern

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 3/1

Page 4: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Simple Descriptive Statistics

Mean center of a point pattern:

� point with coordinates s = (x , y):

x =∑n

i=1 wixi

∑ni=1 wi

and y =∑n

i=1 wiyi

∑ni=1 wi

� center of point pattern, or point with average x and y -coordinates

Median center of a point pattern:

� both of the following two centers are called median centers,although they are essentially di�erent (confusing!)

I the intersection between the median of the x and the y coordinates

I center for minimum distance: sc ∈ {s1, . . . , sn}s.t.minn

∑i=1|s i − sc |

� the �rst type of median center is not unique, and there is no closedform for the second type

� p-median problem (a tppical problem in spatial optimization ): the

problem of locating p �facilities� relative to a set of �customers� such that the sum of

the shortest demand weighted distance between �customers� and �facilities� is minimizedGuofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 4/1

Page 5: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Simple Descriptive Statistics

Changes of population center (year 1790-2000):

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 5/1

Page 6: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Descriptive Statistics

Standard distance of a point pattern:

� average squared deviations of x and y coordinates from theirrespective mean:

dstd =

√∑n

i=1(xi − x)2 + ∑ni=1(yi − y)2

n− 2

� related to standard deviation of coordinates, a summary circle(centered at s with radius dstd ) of a point pattern

Standard devitional ellipse:

� Taking directional e�ects into account for anisotropy cases

� Please refer to Levine and Associates, 2004 for calculations

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 6/1

Page 7: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Descriptive Statistics

Examples:

605000 610000 615000 620000

4825

000

4830

000

4835

000

4840

000

Easting (m)

Nor

thin

g (m

)

SDD

600000 605000 610000 615000 62000048

2000

048

2500

048

3000

048

3500

048

4000

0

Easting (m)

Nor

thin

g (m

)

SDE

Remarks:

� indicates overall shape and center of point pattern

� do not su�ce to fully specify a spatial point pattern

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 7/1

Page 8: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Point Pattern Analysis Methods

1st order (i.e., intensity): absolute location of events on map:

� Quadrat methods

� Density Estimation (KDE)

� Moran's I and Geary's C

2nd order (i.e., interactions): interaction of events:

� Nearest neighbor distance

� Distance functions G, K, F, L

� Getis-Ord Gi* and Anselin local Moran's I

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 8/1

Page 9: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Quadrat methods

Consider a point pattern with n events within a study region A of area |A|

Global intensity:

λ =n

|A| =#of events withinA

|A|

Local intensity via quadrats

1. partition A into L sub-regions Al , l = 1, . . . , L of equal area |Al |(also called quadrats)

2. count number of events n(Al ) in each sub-region Al

3. convert sample counts into estimated intensity rates as:

λ(Al ) =n(Al )

|Al |

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 9/1

Page 10: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Quadrat methods

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337 608 162 73 105 268

422 49 17 52 128 146

231 134 92 406 310 64

� estimated rates λ(Al ) over set of quadrats

� reveal large-scale patterns in intensity variation over A

� larger quadrats yield smoother intensity maps; smaller quadrats yield`spiky' intensity maps

� size, origin, and shape of quadrats is critical (recall: MAUP)

� only �rst-order e�ects are captured

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 10/1

Page 11: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Dependence of intensity on a covariate (Inhomogeneous cases)

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quadrat based on reclass−ed slope

271

9841028

1321

0.00 0.05 0.10 0.15 0.20 0.25 0.30

intensity vs. slope

ρ(Z)ρhi(Z)ρlo(Z)

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 11/1

Page 12: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Kernel Density Estimation

Procedure of Kernel Density Estimation (KDE)

1. de�ne a kernel K (s; r) of radius (or bandwidth) r centered at anyarbitrary location s

2. estimate local intensity at s as:

λ(s) =1

n

n

∑i=1

K (s i − s; r)

3. repeat estimation for all points s in the study region to create adensity map

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 12/1

Page 13: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Kernel Density Estimation

Example for the previous dataset:

den

0.00

50.

015

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Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 13/1

Page 14: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Kernel Density Estimation

Example with 2km bandwidth

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 14/1

Page 15: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Kernel Density Estimation

Example with 10km bandwidth

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 15/1

Page 16: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Kernel Density Estimation

Example with 40km bandwidth

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 16/1

Page 17: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Kernel Density Estimation

Comments

� Choice of kernel function is not critical (Diggle, 1985)

� Choice of bandwidth, or degree of smoothing critical:I Small bandwidth → spiky resultsI Large bandwidth → loss of detail

� Multi-scale analyses can use these bandwidth characteristics toinvestigate both broad trends and localized variation

� How to choose bandwidth: choose the degree of smoothingsubjectively, by eye, or by formula (Diggle)

� could de�ne local bandwidth based on function of presence of eventsin neighborhood of s (i.e., adaptive kernel estimation)

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 17/1

Page 18: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Distance-based Descriptors of Point Patterns

� Distances: accessing second order e�ectsI Event-to-event distance: distance dij between event at arbitrary

location si and another event at another arbitrary location sj :

dij =√(xi − xj )2 + (yi − yj )2

I Point-to-event distance: distance dpj between a randomly chosenpoint at location sp and an event at location sj :

dpj =√(xp − xj )2 + (yi − yj )2

I Event-to-nearest-neighbour distance: distance dmin(si ) between anevent at location si and its nearest neighbor event:

dmin(si ) = min{dij , j 6= i , j = 1, . . . ,n}

I Point-to-nearest-neighbour distance (i.e., empty space distance):distance dmin(sp) between a randomly chosen point at location spand its nearest neighbor event:

dmin(sp) = min{dpj , j = 1, . . . ,n}

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 18/1

Page 19: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Event-to-Nearest-Neighbor Distances

606000 610000 614000 61800048

2800

048

3200

048

3600

048

4000

0

X

Y

1

2

3

4

5

1 2 3 4 5

1 0.00 11947.70 16042.65 3481.22 10742.982 11947.70 0.00 5126.79 15219.58 1599.073 16042.65 5126.79 0.00 19481.59 6720.594 3481.22 15219.58 19481.59 0.00 13913.705 10742.98 1599.07 6720.59 13913.70 0.00

Table : Euclidean distance matrix

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 19/1

Page 20: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Event-to-Nearest-Neighbor Distances

Nearest neighour distances

82 clustering points

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82 clustering points

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cluster

nndist(cluster)

Fre

quen

cy

0.0 0.1 0.2 0.3 0.4

05

1015

2025

3035

uniform

nndist(unif)

Fre

quen

cy

0.00 0.05 0.10 0.150

24

68

1012

� Mean nearest neighbour distance: Average of all dmin(si ) values

dmin =1

n

n

∑i=1

dmin(si )

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 20/1

Page 21: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

The G function

� De�nition: nearest neighbour distance function, i.e., proportion ofevent-to-nearest-neighbor distances dmin(si ) no greater than givendistance cuto� d , estimated as:

G (d) =#{dmin(si ) < d , i = 1, . . . , n}

n

� alternative de�nition: cumulative distribution function (CDF) of alln event-to-nearest-neighbor distances; instead of computing averagedmin of dmin values, compute their CDF

� the G function provides information on event proximity

� example for previous clustering point pattern:Histogram of dm[which(dm > 0)]

dm[which(dm > 0)]

Fre

quen

cy

0.0 0.1 0.2 0.3 0.4

05

1015

2025

3035

0.0 0.1 0.2 0.3 0.4

0.0

0.2

0.4

0.6

0.8

1.0

dm[which(dm > 0)]

Pro

port

ion

<=

x

n:82 m:0

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 21/1

Page 22: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Examples of G function

82 clustering points

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82 clustering points

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0.00 0.05 0.10 0.15 0.20

0.0

0.2

0.4

0.6

0.8

1.0

ghatcluster

r

Gra

w(r)

0.00 0.05 0.10 0.15 0.20

0.0

0.2

0.4

0.6

0.8

1.0

ghatunif

r

Gra

w(r)

Expected plot:

� for clustered events, G (d) rises sharply at short distances, and thenlevels o� at larger d -values

� for randomly-spaced events, G (d) rises gradually up to the distanceat which most events are spaced, and then increases sharply

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 22/1

Page 23: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

The F function

De�nition

� proportion of point-to-nearest-neighbor distances (i.e., empty spacedistances) dmin(sp) no greater than given distance cuto� d ,estimated as:

F (d) =#{dmin(sp) < d , p = 1, . . . ,m}

m

� alternative de�nition: cumulative distribution function (CDF) of allm point-to-nearest-neighbor distances

� the F function provides information on event proximity to voids

� Examples for previous clustering point pattern:

82 clustering points

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empty space distances

0.05

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0.15

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0.00 0.05 0.10 0.15 0.20

0.0

0.2

0.4

0.6

0.8

fhatcluster

r

Fra

w(r)

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 23/1

Page 24: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Examples of F function

82 clustering points

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82 uniform points

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0.00 0.05 0.10 0.15 0.20

0.0

0.2

0.4

0.6

0.8

fhatcluster

r

Fra

w(r)

0.00 0.02 0.04 0.06 0.08

0.0

0.2

0.4

0.6

0.8

fhatunif

r

Fra

w(r)

Expected plot:

� for clustered events, F (d) rises sharply at short distances, and thenlevels o� at larger d -values

� for randomly-spaced events, F (d) rises rapidly up to the distance atwhich most events are spaced, and then levels o� (there are morenearest neighbors at small distances from randomly placed points)

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 24/1

Page 25: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Comparing G and F functions

82 clustering points

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82 uniform points

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0.00 0.05 0.10 0.15 0.20

0.0

0.2

0.4

0.6

0.8

Ghat vs. Fhat

r

Fra

w(r)

0.00 0.02 0.04 0.06 0.08

0.0

0.2

0.4

0.6

0.8

Ghat vs. Fhat

rF

raw(r)

Expected plot:

� for clustered events, G (d) rises faster

� for randomly-spaced events, F (d) tends to be close to G (d)

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 25/1

Page 26: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

The K function

Working with pair-wise distances&looking beyond nearest neighours

Concept

1. construct set of concentric circles (of increasing radius d) aroundeach event

2. count number of events in each distance �band�

3. cumulative number of events up to radius d around all eventsbecomes the sample K function K (d)

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 26/1

Page 27: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

The K function

Working with pair-wise distances&looking beyond nearest neighours

� Formal de�nition:

K (d) =1

λ

#{dij ≤ d , i , j = 1, . . . , n}n

=|A|n

#{dij ≤ d , i , j = 1, . . . , n}n

= |A|(proportion of event-to-event distance ≤ d)

� In other words, the K (d) is the sample cumulative distributionfunction (CDF) of all n2 event-to-event distances, scaled by |A|

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 27/1

Page 28: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Examples of Event-to-Event Distance Histogram and CDFs

82 clustering points

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82 uniform points

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cluster histogram

pairdist(cluster)

Fre

quen

cy

0.0 0.2 0.4 0.6 0.8 1.0

010

020

030

040

050

060

070

0

uniform histogram

pairdist(unif)F

requ

ency

0.0 0.2 0.4 0.6 0.8 1.0 1.2

010

020

030

040

050

0

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

cluster CDF

pairdist(cluster)

Pro

port

ion

<=

x

n:6724 m:0

0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.0

0.2

0.4

0.6

0.8

1.0

uniform CDF

pairdist(unif)

Pro

port

ion

<=

x

n:6724 m:0

� for clustered events, there are multiple bumps in the CDF of E2Edistances due to the grouping of events in space

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 28/1

Page 29: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Examples of K functions

82 clustering points

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82 uniform points

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0.00 0.05 0.10 0.15 0.20 0.25

0.00

0.05

0.10

0.15

0.20

0.25

Khatcluster

r

Kun

(r)

0.00 0.05 0.10 0.15 0.20 0.25

0.00

0.05

0.10

0.15

Khatunif

r

Kun

(r)

� the sample K function K (d) is monotonically increasing and is ascaled (by area |A|) version of the CDF of E2E distances

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 29/1

Page 30: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Recap

Spatial point patterns

� set of n point locations with recorded �events�

Describing the �rst-order e�ect

� overal intensity

� local intensity (quadrat count and kernel density estimation)

Describing the second-order e�ect

� nearest neighbour distancesI the G function

� empty space distancesI the F function

� pair-wise distancesI the K function

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 30/1

Page 31: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Caveats

Caveats:

� theoretical G, F, K functions are de�ned and estimated under theassumption that the point process is stationary (homogeneous)

� these summary functions do not completely characterise the process

� if the process is not stationary, deviations between the empirical andtheoretical functions (e.g. K and K) are not necessarily evidence ofinterpoint interaction, since they may also be attributable tovariations in intensity

Examplecaveat #2

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0.00 0.05 0.10 0.15 0.20 0.25

0.00

0.05

0.10

0.15

0.20

caveat #2

r

K(r)

Kiso(r)Ktrans(r)Kbord(r)Kpois(r)

caveat #3

● ●

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0.00 0.05 0.10 0.15 0.20 0.25

0.00

0.05

0.10

0.15

0.20

0.25

caveat #3

r

K(r)

Kiso(r)Ktrans(r)Kbord(r)Kpois(r)

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 31/1

Page 32: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Descriptive vs Statistical Point Pattern Analysis

Descriptive analysis:

� set of quantitative (and graphical) tools for characterizing spatialpoint patterns

� di�erent tools are appropriate for investigating �rst- or second-ordere�ects (e.g., kernel density estimation versus sample G function)

� can shed light onto whether points are clustered or evenlydistributed in space

Limitation:

� no assessment of how clustered or how evenly-spaced is an observedpoint pattern

� no yardstick against which to compare observed values (or graph) ofresults

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 32/1

Page 33: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Descriptive vs Statistical Point Pattern Analysis

Statistical analysis:

� assessment of whether an observed point pattern can be regarded asone (out of many) realizations from a particular spatial process

� measures of con�dence with which the above assessment can bemade (how likely is that the observed pattern is a realization of aparticular spatial process)

Are daisies randomly distributed in your garden?

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 33/1

Page 34: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Complete Spatial Randomness (CSR)

Complete Spatial Randomness (CSR)

� yardstick, reference model that observed point patterns could becompared with, i.e., null hypothesis

� = homogeneous (uniform) Poisson point process

� basic properties:I the number of points falling in any region A has a Poisson

distribution with mean λ|A|I given that there are n points inside region A, the locations of these

points are i.i.d. and uniformly distributed inside AI the contents of two disjoint regions A and B are independent

Example:csr example #1

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csr example #2

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Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 34/1

Page 35: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Quadrat counting test for CSR

Quadrat counting test

� partition study area A into L sub-regions (quadrats), A1, . . . ,AL

� count number of events n(Al ) in each sub-region Al

� Under the null hypothesis of CSR, the n(Al ) are i.i.d. Poissonrandom variables with the same expected value

� The Pearson χ2 goodness-of-�t test can be usedI test statistics: Pearson residual ∑l ε(Al )

2

ε(Al ) =n(Al )− µ(Al )√

µ(Al ),

where µ(Al ) indicates the expected number of events in AlI ∑l=1 ε(Al )

2 is assumed to follow χ2 distribution

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 35/1

Page 36: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Quadrat counting test for CSR

Exampleunif

●●

14 18 9

14 13 14

13.7 13.7 13.7

13.7 13.7 13.7

0.09 1.2 −1.3

0.09 −0.18 0.09

� three values indicate the number of observations, CSR-expectednumber of observations, and the Pearson residuals

� p-value = 0.617

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 36/1

Page 37: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Nearest Neighbour Index (NNI) test under CSR

Nearest neighbour index

� Compares the mean of the distance observed between each pointand its nearest neighbor (dmin) and the expected mean distanceunder CSR E (dmin)

NNI =dmin

E (dmin)

� Under CSR, we have:

E (dmin) =1

2√

λ

σ(dmin) =0.26136√n2/A

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 37/1

Page 38: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Nearest Neighbour Index (NNI) test under CSR

Nearest neighbour index test

� Test statistics:

z =dmin − E (dmin)

σ(dmin),

� z is assumed to follow Gaussian distribution, thus, if z < −1, 96 orz > 1.96, we are 95% con�dent that the distribution is notrandomly distributed

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 38/1

Page 39: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

The G Function under CSR

� The G function is a function of nearest-neighbour distances

� For a homogeneous Poisson point process of intensity λ, thenearest-neighour distance distribution (the G function) is known tobe:

G (d) = 1− exp{−λπd2}

Example

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0 5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

Gtest

r (metres)

G(r)

Graw(r)Gpois(r)

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 39/1

Page 40: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

The F Function under CSR

� The F function is a function of empty space distances� For a homogeneous Poisson point process of intensity λ, the emptyspace distance distribution (the F function)is known to be:

F (d) = 1− exp{−λπd2}

� Equivalent to the G function� Intuitively, because points (events) of the Poisson process areindependent of each other, the knowledge that a random point is aevent of a point pattern does not a�ect any other event of theprocess

Example

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Fraw(r)Fpois(r)

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 40/1

Page 41: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

The K Function under CSR

� The K function is a function of pair-wise distances� For a homogeneous Poisson point process of intensity λ, thepair-wise distance distribution (the K function) is known to be:

K (d) = πd2

� A commonly-used transformation of K is the L-function:

L(d) =

√K (d)

π= d

Example

bei

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Ktest

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K(r)

Kun(r)Kpois(r)

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012

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Ltest

r (metres)

L(r)

Lun(r)Lpois(r)

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 41/1

Page 42: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Monte Carlo test

� because of random variability, we will never obtain perfectagreement between sample functions (say the K function) withtheoretical functions (the theoretical K functions), even with acompletely random pattern

Example

0.00 0.05 0.10 0.15 0.20 0.25

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0.10

0.12

0.14

Kest(rpoispp(50), correction = "none")

r

Kun

(r)

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 42/1

Page 43: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Monte Carlo test

� A Monte Carlo test is a test based on simulations from the nullhypothesis

� Basic procedures:I generate M independent simulations of CSR inside the study region AI compute the estimated K functions for each of these realisations, sayK (j)(r) for j = 1, . . . ,M

I obtain the pointwise upper and lower envelopes of these simulatedcurves

I not a con�dence interval

Example

0.00 0.05 0.10 0.15 0.20 0.25

0.00

0.05

0.10

0.15

0.20

pointwise envelopes

r

K(r)

Kobs(r)Ktheo(r)Khi(r)Klo(r)

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 43/1

Page 44: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Recap

Statitsical analysis of spatial point patterns:

� allows to quantify departure of results obtained via exploratory tools,e.g., G (d), from expected such results derived under speci�c nullhypotheses, here CSR hypothesis

� can be used to assess to what extent observed point patterns can beregarded as realizations from a particular spatial process (here CSR)

� Same concepts can be applied for hypothesis of other types of pointprocesses (e.g., Poisson cluster process, Cox process)

Sampling distribution of a test statistics

� lies at the heart of any statistical hypothesis testing procedure, andis tied to a particular null hypothesis

� simulation and analytical derivations are two alternative ways ofcomputing such sampling distributions (the latter being increasinglyreplaced by the former)

Edge E�ectsGuofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 44/1

Page 45: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Recap

Scale e�ects

� Wolf pack example

� Nearest neighour distance (NN distance, G,F functions) vs K function

Edge e�ects

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 45/1

Page 46: GIST 4302: Spatial Analysis and Modeling - Point …...Spatial Point Patterns Further issues: analysis of point patterns over large areas should take into account distance distortions

Recap

Extended into line processes

� Line density

Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 46/1