gist 4302: spatial analysis and modeling - point …...spatial point patterns further issues:...
TRANSCRIPT
GIST 4302: Spatial Analysis and ModelingPoint Pattern Analysis
Guofeng Cao
www.myweb.ttu.edu/gucao
Department of Geosciences
Texas Tech University
Fall 2014
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
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
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
Simple Descriptive Statistics
Changes of population center (year 1790-2000):
Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 5/1
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
Descriptive Statistics
Examples:
605000 610000 615000 620000
4825
000
4830
000
4835
000
4840
000
Easting (m)
Nor
thin
g (m
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SDD
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600000 605000 610000 615000 62000048
2000
048
2500
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3000
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3500
048
4000
0
Easting (m)
Nor
thin
g (m
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SDE
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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
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
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
Quadrat methods
bei
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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
Dependence of intensity on a covariate (Inhomogeneous cases)
bei
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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
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
Kernel Density Estimation
Example for the previous dataset:
den
0.00
50.
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y
den
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Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 13/1
Kernel Density Estimation
Example with 2km bandwidth
Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 14/1
Kernel Density Estimation
Example with 10km bandwidth
Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 15/1
Kernel Density Estimation
Example with 40km bandwidth
Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 16/1
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
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
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
Event-to-Nearest-Neighbor Distances
Nearest neighour distances
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
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
Examples of G function
82 clustering points
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0.00 0.05 0.10 0.15 0.20
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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
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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0.05
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0.25
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0.4
0.6
0.8
fhatcluster
r
Fra
w(r)
Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 23/1
Examples of F function
82 clustering points
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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
Comparing G and F functions
82 clustering points
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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
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
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
Examples of Event-to-Event Distance Histogram and CDFs
82 clustering points
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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
Examples of K functions
82 clustering points
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0.00 0.05 0.10 0.15 0.20 0.25
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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
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
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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caveat #2
r
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Kiso(r)Ktrans(r)Kbord(r)Kpois(r)
caveat #3
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caveat #3
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Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 31/1
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
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
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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Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 34/1
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
Quadrat counting test for CSR
Exampleunif
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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
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
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
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
bei
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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
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
bei
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0 5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
Ftest
r (metres)
F(r)
Fraw(r)Fpois(r)
Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 40/1
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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0 20 40 60 80 100 120
010
000
2000
030
000
4000
050
000
Ktest
r (metres)
K(r)
Kun(r)Kpois(r)
0 20 40 60 80 100 120
020
4060
8010
012
0
Ltest
r (metres)
L(r)
Lun(r)Lpois(r)
Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 41/1
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
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Kest(rpoispp(50), correction = "none")
r
Kun
(r)
Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 42/1
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
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
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
Recap
Extended into line processes
� Line density
Guofeng Cao, Texas Tech GIST4302, Point Pattern Analysis 46/1