Sébastien Ollier , Sandrine Pavoine and Pierre...
Transcript of Sébastien Ollier , Sandrine Pavoine and Pierre...
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Comparing and classifyingmulti-scale spatial patterns
Sébastien Ollier*, Sandrine Pavoine and Pierre Couteron
Laboratoire de Biométrie et Biologie EvolutiveCNRS, UMR 5558,
Université Claude Bernard, Lyon1FRANCE
90th Annual Meeting (ESA), August 7-12 2005, Montreal, Canada
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Introduction• Remotely sensed data to understand vegetation/land surface interface
• One sampling unit
PATTERN ANALYSIS
2-D
AutocorelogramFractal dimension
VariogramPeriodogramScalogram...
1-D
VARI
AN
CE
SCALES
kV
k• Many sampling units
1-D
2-D
How to compare many sampling unitson the basis of their spatial patterns?
How to assess the relative importanceof scales in a set of sampling units ?
DALE et al. (2002) Ecography
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I. Pattern quantification1. Two Term Local Variance2. Mean Block Size Variance3. Spectral Analysis
II. Pattern ordination1. Scale standardization2. Ordination by a centred PCA
III.Application to laser profiles1. Data set2. Results
HILL (1973) J. Ecol.
RIPLEY (1978) J. Ecol.
GREIG-SMITH (1952)Ann. bot.
OLLIER et al. (2003)Rem. Sens. Env.
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I. Pattern quantification
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HILL (1973) J. Ecol.
( ) ( ) ( )
( )
2 2 21 1 2 2 3 3 4
1
22 1 2 3 4
2
1
1
ttlv x x x x x xS
ttlv x x x xS
= − + − + −
= + − −
PATTERN ANALYSIS( )1 2 3 4, , ,t x x x x=x
1 1b =
2 2b =
1 4n =
2 2n =
Scalogram
1. Two Term Local Variance
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( )
( )
1 1 2 3 4
2 1 2 2 3 3 4
, , ,
, ,
t
t
x x x x
x x x x x x
=
= + + +
x
x
k k=x H x
1 4
2
1 1 0 00 1 1 00 0 1 1
=
=
H Id
H1
1. Two Term Local Variance
HILL (1973) J. Ecol.
( ) ( ) ( )
( )
2 2 21 1 2 2 3 3 4
1
22 1 2 3 4
2
1
1
ttlv x x x x x xS
ttlv x x x xS
= − + − + −
= + − −
PATTERN ANALYSIS( )1 2 3 4, , ,t x x x x=x
1 1b =
2 2b =
1 4n =
2 2n =
Scalogram
OLLIER et al. (2003)Rem. Sens. Env.
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1 1b =
1
0 1 0 01 0 1 00 1 0 10 0 1 0
M
=
2 2b =
2
0 0 10 0 01 0 0
M =
2
1
1 0 0 00 2 0 00 0 2 00 0 0 1
N
=
2
1 0 00 0 00 0 1
N =
1 2x x+ 2 3x x+ 3 4x x+
1 2 3 4x x x x
1. Two Term Local Variance
( )
( )
1 1 2 3 4
2 1 2 2 3 3 4
, , ,
, ,
t
t
x x x x
x x x x x x
=
= + + +
x
x
k k=x H x
1 4
2
1 1 0 00 1 1 00 0 1 1
=
=
H Id
H1
HILL (1973) J. Ecol.
( ) ( ) ( )
( )
2 2 21 1 2 2 3 3 4
1
22 1 2 3 4
2
1
1
ttlv x x x x x xS
ttlv x x x xS
= − + − + −
= + − −
PATTERN ANALYSIS( )1 2 3 4, , ,t x x x x=x
1 1b =
2 2b =
1 4n =
2 2n =
Scalogram
OLLIER et al. (2003)Rem. Sens. Env.
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3( )t
k k k kk
k
ttlvS−
=x N M x
1. Two Term Local Variance
( )
( )
1 1 2 3 4
2 1 2 2 3 3 4
, , ,
, ,
t
t
x x x x
x x x x x x
=
= + + +
x
x
k k=x H x
1 4
2
1 1 0 00 1 1 00 0 1 1
=
=
H Id
H1
HILL (1973) J. Ecol.
( ) ( ) ( )
( )
2 2 21 1 2 2 3 3 4
1
22 1 2 3 4
2
1
1
ttlv x x x x x xS
ttlv x x x xS
= − + − + −
= + − −
PATTERN ANALYSIS( )1 2 3 4, , ,t x x x x=x
1 1b =
2 2b =
1 4n =
2 2n =
Scalogram
OLLIER et al. (2003)Rem. Sens. Env.
1 1b =
1
0 1 0 01 0 1 00 1 0 10 0 1 0
M
=
2 2b =
2
0 0 10 0 01 0 0
M =
2
1
1 0 0 00 2 0 00 0 2 00 0 0 1
N
=
2
1 0 00 0 00 0 1
N =
1 2x x+ 2 3x x+ 3 4x x+
1 2 3 4x x x x
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1 1b =
2 2b = 1 2x x+ 3 4x x+
1 2 3 4x x x x
( )
ttlv witht
kk
kt
k k k k k
S=
= −
x A x
A H N M H
1. Two Term Local Variance
HILL (1973) J. Ecol.
( ) ( ) ( )
( )
2 2 21 1 2 2 3 3 4
1
22 1 2 3 4
2
1
1
ttlv x x x x x xS
ttlv x x x xS
= − + − + −
= + − −
PATTERN ANALYSIS( )1 2 3 4, , ,t x x x x=x
1 1b =
2 2b =
1 4n =
2 2n =
Scalogram
OLLIER et al. (2003)Rem. Sens. Env.
3( )t
k k k kk
k
ttlvS−
=x N M x
( )
( )
1 1 2 3 4
2 1 2 2 3 3 4
, , ,
, ,
t
t
x x x x
x x x x x x
=
= + + +
x
x
k k=x H x
1 4
2
1 1 0 00 1 1 00 0 1 1
=
=
H Id
H1
2
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2. Mean Block Size Variance
GREIG-SMITH (1952)Ann. bot.
( )1 2 3 4, , ,t x x x x=x
1 1b =
2 2b =
3 4b =
1 4n =
2 2n =
3 1n =
( ) ( )
( )
2 21_ 2 1 2 3 4
1
22_ 4 1 2 3 4
2
1 1 12 2
1 14
msbs x x x xS
msbs x x x xS
= − + −
= + − −
PATTERN ANALYSIS
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( )
( )
( )
1 1 2 3 4
2 1 2 1 2 3 4 3 4
3 1 2 3 4 1 2 3 4
, , ,
, , ,
, ... ,
t
t
t
x x x x
x x x x x x x x
x x x x x x x x
=
= + + + +
= + + + + + +
x
x
x
tk k=x H x1
_ 1 1 11
1 1k k k k k k
k kb b+ + ++
= −y H x H x
2
1 2
2 1
3 4
4 3
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x xx xx xx x
− − − −
1 2 3 4
1 2 3 4
3 4 1 2
3 4 1 2
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x x x xx x x xx x x xx x x x
+ − − + − − + − − + − −
1 4
2
3 4 4
1 1 0 01 1 0 00 0 1 10 0 1 1t
=
=
=
H Id
H
H 1 1
1_ 2y 2_ 4y
3_ 1 _ 1 _ 1t
k k k k k kmsbs + + += y y
2. Mean Block Size Variance
GREIG-SMITH (1952)Ann. bot.
( )1 2 3 4, , ,t x x x x=x
1 1b =
2 2b =
3 4b =
1 4n =
2 2n =
3 1n =
( ) ( )
( )
2 21_ 2 1 2 3 4
1
22_ 4 1 2 3 4
2
1 1 12 2
1 14
msbs x x x xS
msbs x x x xS
= − + −
= + − −
PATTERN ANALYSIS
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_ 1_ 1
1 1 1 11 1
msbs with
1 1 1 1
tk k
k kk
t t t tk k k k k k k k k
k k k k
S
b b b b
++
+ + + ++ +
=
= − −
x A x
A H H H H H H H H
( )
( )
( )
1 1 2 3 4
2 1 2 1 2 3 4 3 4
3 1 2 3 4 1 2 3 4
, , ,
, , ,
, ... ,
t
t
t
x x x x
x x x x x x x x
x x x x x x x x
=
= + + + +
= + + + + + +
x
x
x
tk k=x H x1
_ 1 1 11
1 1k k k k k k
k kb b+ + ++
= −y H x H x
2
1 2
2 1
3 4
4 3
12
x xx xx xx x
− − − −
1 2 3 4
1 2 3 4
3 4 1 2
3 4 1 2
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x x x xx x x xx x x xx x x x
+ − − + − − + − − + − −
1_ 2y 2_ 4y
3_ 1 _ 1 _ 1t
k k k k k kmsbs + + += y y
2. Mean Block Size Variance
GREIG-SMITH (1952)Ann. bot.
( )1 2 3 4, , ,t x x x x=x
1 1b =
2 2b =
3 4b =
1 4n =
2 2n =
3 1n =
( ) ( )
( )
2 21_ 2 1 2 3 4
1
22_ 4 1 2 3 4
2
1 1 12 2
1 14
msbs x x x xS
msbs x x x xS
= − + −
= + − −
PATTERN ANALYSIS
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3.Spectral analysis
RIPLEY (1978) J. Ecol.
( )1 2 3 4, , ,t x x x x=x
( ) ( ) ( )
( ) ( )
2 2
1 1 11 11
2
2 212
1 1 cos sin
1 1 cos
n n
i ii i
n
ii
I x i x iS n
I x iS n
ω ω ω
ω ω
= =
=
= +
=
∑ ∑
∑
1k =
1k =1
2nπω =
2k =2ω π=
Periodogramcos
cos
sin
1c
1s
2c
PATTERN ANALYSIS
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( )
( )
I with
1
tk
kk
t tk k k kk
S
n
ω =
= +
x A x
A c c s s
3.Spectral analysis
RIPLEY (1978) J. Ecol.
( )1 2 3 4, , ,t x x x x=x
( ) ( ) ( )
( ) ( )
2 2
1 1 11 11
2
2 212
1 1 cos sin
1 1 cos
n n
i ii i
n
ii
I x i x iS n
I x iS n
ω ω ω
ω ω
= =
=
= +
=
∑ ∑
∑
1k =
1k =1
2nπω =
2k =2ω π=
Periodogramcos
cos
sin
1c
1s
2c
PATTERN ANALYSIS
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Pattern quantificationMean square block size analaysis
Two term local varance
Spectral analysis
1-D
VARI
AN
CE
SCALES
kV
k
1-D
...
• Many sampling units
• One sampling unit Vt
kk
kS=
x A x
TABLE Z
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How to choose the scaling coeficient ?
Pattern quantificationMean square block size analaysis
Two term local varance
Spectral analysis
1-D
VARI
AN
CE
SCALES
kV
k
1-D
...
• Many sampling units
• One sampling unit Vt
kk
kS=
x A x
TABLE Z
kS
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II. Pattern ordination
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1. Scale standardization
EUCLIDEAN STANDARDIZATION
( )k kS trace A=
• Two term local variance
161, 2, 4,8
nk==
kS
1k = 2 4 8
30 52 72 16
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1. Scale standardization
EUCLIDEAN STANDARDIZATION SPECTRAL STANDARDIZATION
( )k kS trace A= ( )1k kS Aλ=
• Two term local variance
161, 2, 4,8
nk==
kS
1k = 2 4 8
30 52 72 16 kS
1k = 2 4 84 9 26 16
firsteigenvectors
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1. Scale standardization
EUCLIDEAN STANDARDIZATION SPECTRAL STANDARDIZATION
( )k kS trace A= ( )1k kS Aλ=
• Two term local variance
161, 2, 4,8
nk==
kS
1k = 2 4 8
30 52 72 16 kS
1k = 2 4 84 9 26 16
• Mean block size variance
kS 8 4 2 1
• Spectral analysis
kS 2 2 2 1
kS 1 1 1 1
kS 1 1 1 1
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2.Ordination by a centred PCA
SCALES
1-D
...
TABLE Z
Vk values
TRA
NSE
CTS Centered
PrincipalComponentAnalysis
1
p
q...
...• Eigenvalues
• Coordinates onPrincipal Components
• Coordinates onPrincipal Axes
Package ade4 : http://pbil.univ-lyon1.fr/ADE-4Software R : http://cran.univ-lyon1.fr/R
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III. Application
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1. Data set
OLLIER et al. (2003) Rem. Sens. Env.
N
E
S
O
N
E
S
O
The study site
2.5 km
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1. Data set
N
E
S
O
N
E
S
O2.5 km
The study site
• ALLUVIAL PLAINSvery simple
and flat relief
• COMPLEX RELIEF FORMS< 60 m
gently slopes
• COMPLEX RELIEF FORMS> 60 m
steep slopesC
A
B
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1. Data set
N
E
S
O
N
E
S
O2.5 km
The study site
• ALLUVIAL PLAINSvery simple
and flat relief
• COMPLEX RELIEF FORMS< 60 m
gently slopes
• COMPLEX RELIEF FORMS> 60 m
steep slopes
A
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1. Data set
N
E
S
O
N
E
S
O2.5 km
The study site
• ALLUVIAL PLAINSvery simple
and flat relief
• COMPLEX RELIEF FORMS< 60 m
gently slopes
• COMPLEX RELIEF FORMS> 60 m
steep slopes
A
B
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1. Data set
N
E
S
O
N
E
S
O2.5 km
The study site
• ALLUVIAL PLAINSvery simple
and flat relief
• COMPLEX RELIEF FORMS< 60 m
gently slopes
• COMPLEX RELIEF FORMS> 60 m
steep slopes
B
A
C
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1. Data set
N
E
S
O
N
E
S
O2.5 km
Laser data
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1. Data set
264 laser transects
N
E
S
O
N
E
S
O2.5 km
264 laser transects (n=64)
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Pattern analysis and ordination
264 scalograms (block size = 1,2,4,8,16,32)
264 laser transects (n=64)
TTLV
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Pattern analysis and ordination
264 scalograms (block size = 1,2,4,8,16,32)
264 laser transects (n=64)
TTLVPCA
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2.Results
• Eigenvalues
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d = 0.2
ttg_1 ttg_2
ttg_4
ttg_8 ttg_16
ttg_32
2.Results
• Eigenvalues
• Principal axis
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d = 0.2
ttg_1 ttg_2
ttg_4
ttg_8 ttg_16
ttg_32
2.Results
• Coordinates on principal components
• Eigenvalues
• Principal axis
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d = 0.2
1 2 3
4
5 ttg_1 ttg_2
ttg_4
ttg_8 ttg_16
ttg_32
2.Results
• FIVE CLUSTERS• Coordinates on principal components
• Eigenvalues
• Principal axis
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d = 0.2
1 2 3
4
5 ttg_1 ttg_2
ttg_4
ttg_8 ttg_16
ttg_32
2.Results
• FIVE CLUSTERS• Coordinates on principal components
• Eigenvalues
• Principal axis
ttg_1 ttg_2 ttg_4 ttg_8 ttg_16 ttg_32
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
ttg_1 ttg_2 ttg_4 ttg_8 ttg_16 ttg_32
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
ttg_1 ttg_2 ttg_4 ttg_8 ttg_16 ttg_32
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
ttg_1 ttg_2 ttg_4 ttg_8 ttg_16 ttg_32
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
ttg_1 ttg_2 ttg_4 ttg_8 ttg_16 ttg_32
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1 23
4
5
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d = 0.2
1 2 3
4
5 ttg_1 ttg_2
ttg_4
ttg_8 ttg_16
ttg_32
ttg_1 ttg_2 ttg_4 ttg_8 ttg_16 ttg_32
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
ttg_1 ttg_2 ttg_4 ttg_8 ttg_16 ttg_32
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
ttg_1 ttg_2 ttg_4 ttg_8 ttg_16 ttg_32
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
ttg_1 ttg_2 ttg_4 ttg_8 ttg_16 ttg_32
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
ttg_1 ttg_2 ttg_4 ttg_8 ttg_16 ttg_32
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1 23
4
5
2.Results
9191248C
234223248B
186752A54321
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Conclusion
• An efficient method to summarizeand classify 1-D spatial patterns
• A very general approach
• Generalization to 2-D sampling unitsCOUTERON et al. (2005) J. Appl. Ecol.