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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
A Mann-Whitney spatial scan statistic forcontinuous data
Lionel Cucala, Christophe Dematteı
August 24th 2010
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Outline
Introduction
1-Potential clusters
2-A Mann-Whitney concentration index
3-Results
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Outline
Introduction
1-Potential clusters
2-A Mann-Whitney concentration index
3-Results
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
A data set to analyze
2000 4000 6000 8000 10000
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long[1:339]
lat[1:
339]
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
A data set to analyze
2000 4000 6000 8000 10000
1800
020
000
2200
024
000
2600
0
long[1:339]
lat[1:
339]
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
A data set to analyze
è 339 dairy farms located in France.
è Somatic individual score (indicator for a disease calledmastitis).
è Marked point process : (Xi ,Ci ), i = 1, · · · , n whereXi ∈ A ⊂ Rd and Ci ∈ R.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
A data set to analyze
è 339 dairy farms located in France.
è Somatic individual score (indicator for a disease calledmastitis).
è Marked point process : (Xi ,Ci ), i = 1, · · · , n whereXi ∈ A ⊂ Rd and Ci ∈ R.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
A data set to analyze
è 339 dairy farms located in France.
è Somatic individual score (indicator for a disease calledmastitis).
è Marked point process : (Xi ,Ci ), i = 1, · · · , n whereXi ∈ A ⊂ Rd and Ci ∈ R.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
A data set to analyze
è 339 dairy farms located in France.
è Somatic individual score (indicator for a disease calledmastitis).
è Marked point process : (Xi ,Ci ), i = 1, · · · , n whereXi ∈ A ⊂ Rd and Ci ∈ R.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Definition :Cluster= geographical area where the continuous variable is higherthan outside.
Question :Are there one or more clusters ? Where ?
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Definition :Cluster= geographical area where the continuous variable is higherthan outside.
Question :Are there one or more clusters ? Where ?
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Definition :Cluster= geographical area where the continuous variable is higherthan outside.
Question :Are there one or more clusters ? Where ?
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
The null hypothesis
è We attempt to reject H0 : (C1, · · · ,CN) independent andidentically distributed.
è Goals : detecting cluster(s) and testing the significance accordingto H0.
Two steps :
• defining the set of potential clusters.
• choosing a concentration index.
Statistic : maximal concentration among potential clusters.
Significance estimated by a Monte-Carlo procedure.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
The null hypothesis
è We attempt to reject H0 : (C1, · · · ,CN) independent andidentically distributed.
è Goals : detecting cluster(s) and testing the significance accordingto H0.
Two steps :
• defining the set of potential clusters.
• choosing a concentration index.
Statistic : maximal concentration among potential clusters.
Significance estimated by a Monte-Carlo procedure.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
The null hypothesis
è We attempt to reject H0 : (C1, · · · ,CN) independent andidentically distributed.
è Goals : detecting cluster(s) and testing the significance accordingto H0.
Two steps :
• defining the set of potential clusters.
• choosing a concentration index.
Statistic : maximal concentration among potential clusters.
Significance estimated by a Monte-Carlo procedure.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
The null hypothesis
è We attempt to reject H0 : (C1, · · · ,CN) independent andidentically distributed.
è Goals : detecting cluster(s) and testing the significance accordingto H0.
Two steps :
• defining the set of potential clusters.
• choosing a concentration index.
Statistic : maximal concentration among potential clusters.
Significance estimated by a Monte-Carlo procedure.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
The null hypothesis
è We attempt to reject H0 : (C1, · · · ,CN) independent andidentically distributed.
è Goals : detecting cluster(s) and testing the significance accordingto H0.
Two steps :
• defining the set of potential clusters.
• choosing a concentration index.
Statistic : maximal concentration among potential clusters.
Significance estimated by a Monte-Carlo procedure.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
The null hypothesis
è We attempt to reject H0 : (C1, · · · ,CN) independent andidentically distributed.
è Goals : detecting cluster(s) and testing the significance accordingto H0.
Two steps :
• defining the set of potential clusters.
• choosing a concentration index.
Statistic : maximal concentration among potential clusters.
Significance estimated by a Monte-Carlo procedure.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
The null hypothesis
è We attempt to reject H0 : (C1, · · · ,CN) independent andidentically distributed.
è Goals : detecting cluster(s) and testing the significance accordingto H0.
Two steps :
• defining the set of potential clusters.
• choosing a concentration index.
Statistic : maximal concentration among potential clusters.
Significance estimated by a Monte-Carlo procedure.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
The null hypothesis
è We attempt to reject H0 : (C1, · · · ,CN) independent andidentically distributed.
è Goals : detecting cluster(s) and testing the significance accordingto H0.
Two steps :
• defining the set of potential clusters.
• choosing a concentration index.
Statistic : maximal concentration among potential clusters.
Significance estimated by a Monte-Carlo procedure.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Outline
Introduction
1-Potential clusters
2-A Mann-Whitney concentration index
3-Results
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Outline
Introduction
1-Potential clusters
2-A Mann-Whitney concentration index
3-Results
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Fixed-shape potential clusters
è Circles centered on one event, another event on the circumference.
è Elliptic clusters, with given orientation and shape (2D only).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Fixed-shape potential clusters
è Circles centered on one event, another event on the circumference.
è Elliptic clusters, with given orientation and shape (2D only).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Fixed-shape potential clusters
è Circles centered on one event, another event on the circumference.
è Elliptic clusters, with given orientation and shape (2D only).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Data-based potential clusters
Graphs G(δ) associated to the point process :
è Vertices : {1, · · · , n}.
è Edges : {(i , j) : d(xi , xj) ≤ δ, 1 ≤ i < n, i < j ≤ n}.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Data-based potential clusters
Graphs G(δ) associated to the point process :
è Vertices : {1, · · · , n}.
è Edges : {(i , j) : d(xi , xj) ≤ δ, 1 ≤ i < n, i < j ≤ n}.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Data-based potential clusters
Graphs G(δ) associated to the point process :
è Vertices : {1, · · · , n}.
è Edges : {(i , j) : d(xi , xj) ≤ δ, 1 ≤ i < n, i < j ≤ n}.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Data-based potential clusters
Graphs G(δ) associated to the point process :
è Vertices : {1, · · · , n}.
è Edges : {(i , j) : d(xi , xj) ≤ δ, 1 ≤ i < n, i < j ≤ n}.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Data-based potential clusters
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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Data-based potential clusters
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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Data-based potential clusters
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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Data-based potential clusters
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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Data-based potential clusters
−0.5 0.0 0.5 1.0 1.5
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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Data-based potential clusters
−0.5 0.0 0.5 1.0 1.5
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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Data-based potential clusters
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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Data-based potential clusters
è Connected component of the edge i in G(δ) : Ni (δ).
è Ai (δ) = {x ∈ A : ∃j ∈ Ni (δ), d(x , xj) ≤ δ}.
è Potential clusters :
C= {Ai (δ) : 1 ≤ i ≤ n, δ ∈ R+}.
è Only n − 1 areas, arbitrarily shaped.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Data-based potential clusters
è Connected component of the edge i in G(δ) : Ni (δ).
è Ai (δ) = {x ∈ A : ∃j ∈ Ni (δ), d(x , xj) ≤ δ}.
è Potential clusters :
C= {Ai (δ) : 1 ≤ i ≤ n, δ ∈ R+}.
è Only n − 1 areas, arbitrarily shaped.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Data-based potential clusters
è Connected component of the edge i in G(δ) : Ni (δ).
è Ai (δ) = {x ∈ A : ∃j ∈ Ni (δ), d(x , xj) ≤ δ}.
è Potential clusters :
C= {Ai (δ) : 1 ≤ i ≤ n, δ ∈ R+}.
è Only n − 1 areas, arbitrarily shaped.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Data-based potential clusters
è Connected component of the edge i in G(δ) : Ni (δ).
è Ai (δ) = {x ∈ A : ∃j ∈ Ni (δ), d(x , xj) ≤ δ}.
è Potential clusters :
C= {Ai (δ) : 1 ≤ i ≤ n, δ ∈ R+}.
è Only n − 1 areas, arbitrarily shaped.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Data-based potential clusters
è Connected component of the edge i in G(δ) : Ni (δ).
è Ai (δ) = {x ∈ A : ∃j ∈ Ni (δ), d(x , xj) ≤ δ}.
è Potential clusters :
C= {Ai (δ) : 1 ≤ i ≤ n, δ ∈ R+}.
è Only n − 1 areas, arbitrarily shaped.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Outline
Introduction
1-Potential clusters
2-A Mann-Whitney concentration index
3-Results
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Outline
Introduction
1-Potential clusters
2-A Mann-Whitney concentration index
3-Results
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Concentration indices
We evaluate the concentration in Z ⊂ A.
è n(Z ) = ]{i : Xi ∈ Z}.
è µ(Z ) = 1n(Z)
∑i :Xi∈Z Ci .
è σ(Z )2 = 1n(Z)
∑i :Xi∈Z
(Ci − µ(Z )
)2.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Concentration indices
We evaluate the concentration in Z ⊂ A.
è n(Z ) = ]{i : Xi ∈ Z}.
è µ(Z ) = 1n(Z)
∑i :Xi∈Z Ci .
è σ(Z )2 = 1n(Z)
∑i :Xi∈Z
(Ci − µ(Z )
)2.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Concentration indices
We evaluate the concentration in Z ⊂ A.
è n(Z ) = ]{i : Xi ∈ Z}.
è µ(Z ) = 1n(Z)
∑i :Xi∈Z Ci .
è σ(Z )2 = 1n(Z)
∑i :Xi∈Z
(Ci − µ(Z )
)2.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Concentration indices
We evaluate the concentration in Z ⊂ A.
è n(Z ) = ]{i : Xi ∈ Z}.
è µ(Z ) = 1n(Z)
∑i :Xi∈Z Ci .
è σ(Z )2 = 1n(Z)
∑i :Xi∈Z
(Ci − µ(Z )
)2.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Concentration indices
We evaluate the concentration in Z ⊂ A.
è n(Z ) = ]{i : Xi ∈ Z}.
è µ(Z ) = 1n(Z)
∑i :Xi∈Z Ci .
è σ(Z )2 = 1n(Z)
∑i :Xi∈Z
(Ci − µ(Z )
)2.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Likelihood-based indices
We test the presence of a cluster in Z ⊂ A.
è H0 : Ci ∼ f0.
è H1,Z : Ci |Xi ∼ fZ 11(Xi ∈ Z ) + fZ 11(Xi ∈ Z ).
Likelihood ratio :
I (Z ) =L1,Z (X1, · · · ,Xn,C1, · · · ,Cn)
L0(X1, · · · ,Xn,C1, · · · ,Cn)11(µ(Z ) > µ(Z )
).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Likelihood-based indices
We test the presence of a cluster in Z ⊂ A.
è H0 : Ci ∼ f0.
è H1,Z : Ci |Xi ∼ fZ 11(Xi ∈ Z ) + fZ 11(Xi ∈ Z ).
Likelihood ratio :
I (Z ) =L1,Z (X1, · · · ,Xn,C1, · · · ,Cn)
L0(X1, · · · ,Xn,C1, · · · ,Cn)11(µ(Z ) > µ(Z )
).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Likelihood-based indices
We test the presence of a cluster in Z ⊂ A.
è H0 : Ci ∼ f0.
è H1,Z : Ci |Xi ∼ fZ 11(Xi ∈ Z ) + fZ 11(Xi ∈ Z ).
Likelihood ratio :
I (Z ) =L1,Z (X1, · · · ,Xn,C1, · · · ,Cn)
L0(X1, · · · ,Xn,C1, · · · ,Cn)11(µ(Z ) > µ(Z )
).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Likelihood-based indices
We test the presence of a cluster in Z ⊂ A.
è H0 : Ci ∼ f0.
è H1,Z : Ci |Xi ∼ fZ 11(Xi ∈ Z ) + fZ 11(Xi ∈ Z ).
Likelihood ratio :
I (Z ) =L1,Z (X1, · · · ,Xn,C1, · · · ,Cn)
L0(X1, · · · ,Xn,C1, · · · ,Cn)11(µ(Z ) > µ(Z )
).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Likelihood-based indices
We test the presence of a cluster in Z ⊂ A.
è H0 : Ci ∼ f0.
è H1,Z : Ci |Xi ∼ fZ 11(Xi ∈ Z ) + fZ 11(Xi ∈ Z ).
Likelihood ratio :
I (Z ) =L1,Z (X1, · · · ,Xn,C1, · · · ,Cn)
L0(X1, · · · ,Xn,C1, · · · ,Cn)11(µ(Z ) > µ(Z )
).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Likelihood-based indices
The Exponential model
è H0 : Ci ∼ E(1/µ(A)
).
è H1,Z : Ci |Xi ∼ E(1/µ(Z ))
11(Xi ∈ Z ) + E(1/µ(Z )
)11(Xi ∈ Z ).
Likelihood ratio :
Iexp(Z ) = −n(Z ) log(µ(Z )
)− n(Z ) log
(µ(Z )
).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Likelihood-based indices
The Exponential model
è H0 : Ci ∼ E(1/µ(A)
).
è H1,Z : Ci |Xi ∼ E(1/µ(Z ))
11(Xi ∈ Z ) + E(1/µ(Z )
)11(Xi ∈ Z ).
Likelihood ratio :
Iexp(Z ) = −n(Z ) log(µ(Z )
)− n(Z ) log
(µ(Z )
).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Likelihood-based indices
The Exponential model
è H0 : Ci ∼ E(1/µ(A)
).
è H1,Z : Ci |Xi ∼ E(1/µ(Z ))
11(Xi ∈ Z ) + E(1/µ(Z )
)11(Xi ∈ Z ).
Likelihood ratio :
Iexp(Z ) = −n(Z ) log(µ(Z )
)− n(Z ) log
(µ(Z )
).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Likelihood-based indices
The Exponential model
è H0 : Ci ∼ E(1/µ(A)
).
è H1,Z : Ci |Xi ∼ E(1/µ(Z ))
11(Xi ∈ Z ) + E(1/µ(Z )
)11(Xi ∈ Z ).
Likelihood ratio :
Iexp(Z ) = −n(Z ) log(µ(Z )
)− n(Z ) log
(µ(Z )
).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Likelihood-based indices
The Exponential model
è H0 : Ci ∼ E(1/µ(A)
).
è H1,Z : Ci |Xi ∼ E(1/µ(Z ))
11(Xi ∈ Z ) + E(1/µ(Z )
)11(Xi ∈ Z ).
Likelihood ratio :
Iexp(Z ) = −n(Z ) log(µ(Z )
)− n(Z ) log
(µ(Z )
).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Likelihood-based indices
The homoscedastic Gaussian model
è H0 : Ci ∼ N(µ(A), σ(A)2
).
è H1,Z : Ci |Xi ∼ N(µ(Z ), σ2
1,Z
)11(Xi ∈ Z )
+N(µ(Z ), σ2
1,Z
)11(Xi ∈ Z )
where σ21,Z = n(Z)σ(Z)2+n(Z)σ(Z)2
n .
Likelihood ratio :
Ihomgau(Z ) =1
σ21,Z
.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Likelihood-based indices
The homoscedastic Gaussian model
è H0 : Ci ∼ N(µ(A), σ(A)2
).
è H1,Z : Ci |Xi ∼ N(µ(Z ), σ2
1,Z
)11(Xi ∈ Z )
+N(µ(Z ), σ2
1,Z
)11(Xi ∈ Z )
where σ21,Z = n(Z)σ(Z)2+n(Z)σ(Z)2
n .
Likelihood ratio :
Ihomgau(Z ) =1
σ21,Z
.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Likelihood-based indices
The homoscedastic Gaussian model
è H0 : Ci ∼ N(µ(A), σ(A)2
).
è H1,Z : Ci |Xi ∼ N(µ(Z ), σ2
1,Z
)11(Xi ∈ Z )
+N(µ(Z ), σ2
1,Z
)11(Xi ∈ Z )
where σ21,Z = n(Z)σ(Z)2+n(Z)σ(Z)2
n .
Likelihood ratio :
Ihomgau(Z ) =1
σ21,Z
.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Likelihood-based indices
The homoscedastic Gaussian model
è H0 : Ci ∼ N(µ(A), σ(A)2
).
è H1,Z : Ci |Xi ∼ N(µ(Z ), σ2
1,Z
)11(Xi ∈ Z )
+N(µ(Z ), σ2
1,Z
)11(Xi ∈ Z )
where σ21,Z = n(Z)σ(Z)2+n(Z)σ(Z)2
n .
Likelihood ratio :
Ihomgau(Z ) =1
σ21,Z
.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Likelihood-based indices
The homoscedastic Gaussian model
è H0 : Ci ∼ N(µ(A), σ(A)2
).
è H1,Z : Ci |Xi ∼ N(µ(Z ), σ2
1,Z
)11(Xi ∈ Z )
+N(µ(Z ), σ2
1,Z
)11(Xi ∈ Z )
where σ21,Z = n(Z)σ(Z)2+n(Z)σ(Z)2
n .
Likelihood ratio :
Ihomgau(Z ) =1
σ21,Z
.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Likelihood-based indices
The homoscedastic Gaussian model
è H0 : Ci ∼ N(µ(A), σ(A)2
).
è H1,Z : Ci |Xi ∼ N(µ(Z ), σ2
1,Z
)11(Xi ∈ Z )
+N(µ(Z ), σ2
1,Z
)11(Xi ∈ Z )
where σ21,Z = n(Z)σ(Z)2+n(Z)σ(Z)2
n .
Likelihood ratio :
Ihomgau(Z ) =1
σ21,Z
.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Likelihood-based indices
The heteroscedastic Gaussian model
è H0 : Ci ∼ N(µ(A), σ(A)2
).
è H1,Z : Ci |Xi ∼ N(µ(Z ), σ(Z )2
)11(Xi ∈ Z )
+N(µ(Z ), σ(Z )2
)11(Xi ∈ Z ).
Likelihood ratio :
Ihetgau(Z ) = −n(Z ) log(σ(Z )2
)− n(Z ) log
(σ(Z )2
).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Likelihood-based indices
The heteroscedastic Gaussian model
è H0 : Ci ∼ N(µ(A), σ(A)2
).
è H1,Z : Ci |Xi ∼ N(µ(Z ), σ(Z )2
)11(Xi ∈ Z )
+N(µ(Z ), σ(Z )2
)11(Xi ∈ Z ).
Likelihood ratio :
Ihetgau(Z ) = −n(Z ) log(σ(Z )2
)− n(Z ) log
(σ(Z )2
).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Likelihood-based indices
The heteroscedastic Gaussian model
è H0 : Ci ∼ N(µ(A), σ(A)2
).
è H1,Z : Ci |Xi ∼ N(µ(Z ), σ(Z )2
)11(Xi ∈ Z )
+N(µ(Z ), σ(Z )2
)11(Xi ∈ Z ).
Likelihood ratio :
Ihetgau(Z ) = −n(Z ) log(σ(Z )2
)− n(Z ) log
(σ(Z )2
).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Likelihood-based indices
The heteroscedastic Gaussian model
è H0 : Ci ∼ N(µ(A), σ(A)2
).
è H1,Z : Ci |Xi ∼ N(µ(Z ), σ(Z )2
)11(Xi ∈ Z )
+N(µ(Z ), σ(Z )2
)11(Xi ∈ Z ).
Likelihood ratio :
Ihetgau(Z ) = −n(Z ) log(σ(Z )2
)− n(Z ) log
(σ(Z )2
).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Likelihood-based indices
The heteroscedastic Gaussian model
è H0 : Ci ∼ N(µ(A), σ(A)2
).
è H1,Z : Ci |Xi ∼ N(µ(Z ), σ(Z )2
)11(Xi ∈ Z )
+N(µ(Z ), σ(Z )2
)11(Xi ∈ Z ).
Likelihood ratio :
Ihetgau(Z ) = −n(Z ) log(σ(Z )2
)− n(Z ) log
(σ(Z )2
).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
No distribution assumption : Mann-Whitney test
è Rj is the rank of Cj among the Ci , 1 ≤ i ≤ n.
è RS(Z ) =∑
i :Xi∈Z Ri .
è Under H0, E(RS(Z )
)= M(Z ) = n(Z)(n+1)
2 .
è Under H0, Var(RS(Z )
)= V (Z ) = n(Z)n(Z)(n+1)
12
è Under H0, RS(Z)−M(Z)√V (Z)
−→dN (0, 1).
Irank(Z ) =RS(Z )−M(Z )√
V (Z ).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
No distribution assumption : Mann-Whitney test
è Rj is the rank of Cj among the Ci , 1 ≤ i ≤ n.
è RS(Z ) =∑
i :Xi∈Z Ri .
è Under H0, E(RS(Z )
)= M(Z ) = n(Z)(n+1)
2 .
è Under H0, Var(RS(Z )
)= V (Z ) = n(Z)n(Z)(n+1)
12
è Under H0, RS(Z)−M(Z)√V (Z)
−→dN (0, 1).
Irank(Z ) =RS(Z )−M(Z )√
V (Z ).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
No distribution assumption : Mann-Whitney test
è Rj is the rank of Cj among the Ci , 1 ≤ i ≤ n.
è RS(Z ) =∑
i :Xi∈Z Ri .
è Under H0, E(RS(Z )
)= M(Z ) = n(Z)(n+1)
2 .
è Under H0, Var(RS(Z )
)= V (Z ) = n(Z)n(Z)(n+1)
12
è Under H0, RS(Z)−M(Z)√V (Z)
−→dN (0, 1).
Irank(Z ) =RS(Z )−M(Z )√
V (Z ).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
No distribution assumption : Mann-Whitney test
è Rj is the rank of Cj among the Ci , 1 ≤ i ≤ n.
è RS(Z ) =∑
i :Xi∈Z Ri .
è Under H0, E(RS(Z )
)= M(Z ) = n(Z)(n+1)
2 .
è Under H0, Var(RS(Z )
)= V (Z ) = n(Z)n(Z)(n+1)
12
è Under H0, RS(Z)−M(Z)√V (Z)
−→dN (0, 1).
Irank(Z ) =RS(Z )−M(Z )√
V (Z ).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
No distribution assumption : Mann-Whitney test
è Rj is the rank of Cj among the Ci , 1 ≤ i ≤ n.
è RS(Z ) =∑
i :Xi∈Z Ri .
è Under H0, E(RS(Z )
)= M(Z ) = n(Z)(n+1)
2 .
è Under H0, Var(RS(Z )
)= V (Z ) = n(Z)n(Z)(n+1)
12
è Under H0, RS(Z)−M(Z)√V (Z)
−→dN (0, 1).
Irank(Z ) =RS(Z )−M(Z )√
V (Z ).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
No distribution assumption : Mann-Whitney test
è Rj is the rank of Cj among the Ci , 1 ≤ i ≤ n.
è RS(Z ) =∑
i :Xi∈Z Ri .
è Under H0, E(RS(Z )
)= M(Z ) = n(Z)(n+1)
2 .
è Under H0, Var(RS(Z )
)= V (Z ) = n(Z)n(Z)(n+1)
12
è Under H0, RS(Z)−M(Z)√V (Z)
−→dN (0, 1).
Irank(Z ) =RS(Z )−M(Z )√
V (Z ).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
No distribution assumption : Mann-Whitney test
è Rj is the rank of Cj among the Ci , 1 ≤ i ≤ n.
è RS(Z ) =∑
i :Xi∈Z Ri .
è Under H0, E(RS(Z )
)= M(Z ) = n(Z)(n+1)
2 .
è Under H0, Var(RS(Z )
)= V (Z ) = n(Z)n(Z)(n+1)
12
è Under H0, RS(Z)−M(Z)√V (Z)
−→dN (0, 1).
Irank(Z ) =RS(Z )−M(Z )√
V (Z ).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
No distribution assumption : Mann-Whitney test
è Rj is the rank of Cj among the Ci , 1 ≤ i ≤ n.
è RS(Z ) =∑
i :Xi∈Z Ri .
è Under H0, E(RS(Z )
)= M(Z ) = n(Z)(n+1)
2 .
è Under H0, Var(RS(Z )
)= V (Z ) = n(Z)n(Z)(n+1)
12
è Under H0, RS(Z)−M(Z)√V (Z)
−→dN (0, 1).
Irank(Z ) =RS(Z )−M(Z )√
V (Z ).
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Outline
Introduction
1-Potential clusters
2-A Mann-Whitney concentration index
3-Results
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Outline
Introduction
1-Potential clusters
2-A Mann-Whitney concentration index
3-Results
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
The farms data set
2000 4000 6000 8000 10000
1800
020
000
2200
024
000
2600
0
long[1:339]
lat[1:
339]
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
The farms data set
2000 4000 6000 8000 10000
1800
020
000
2200
024
000
2600
0
long[1:339]
lat[1:
339]
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
The farms data set
2000 4000 6000 8000 10000
1800
020
000
2200
024
000
2600
0
long[1:339]
lat[1:
339]
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
The farms data set
2000 4000 6000 8000 10000
1800
020
000
2200
024
000
2600
0
long[1:339]
lat[1:
339]
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Astronomical data
Observation of a cubic part of the Universe : (20 Mpc)3 .1 Mpc = 3× 1022 metres .
è Locations of the galaxies : X1, · · · ,Xn.
è Light intensities : C1, · · · ,Cn.
Goal : detecting areas where galaxies are ”redder”.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Astronomical data
Observation of a cubic part of the Universe : (20 Mpc)3 .
1 Mpc = 3× 1022 metres .
è Locations of the galaxies : X1, · · · ,Xn.
è Light intensities : C1, · · · ,Cn.
Goal : detecting areas where galaxies are ”redder”.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Astronomical data
Observation of a cubic part of the Universe : (20 Mpc)3 .1 Mpc = 3× 1022 metres .
è Locations of the galaxies : X1, · · · ,Xn.
è Light intensities : C1, · · · ,Cn.
Goal : detecting areas where galaxies are ”redder”.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Astronomical data
Observation of a cubic part of the Universe : (20 Mpc)3 .1 Mpc = 3× 1022 metres .
è Locations of the galaxies : X1, · · · ,Xn.
è Light intensities : C1, · · · ,Cn.
Goal : detecting areas where galaxies are ”redder”.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Astronomical data
Observation of a cubic part of the Universe : (20 Mpc)3 .1 Mpc = 3× 1022 metres .
è Locations of the galaxies : X1, · · · ,Xn.
è Light intensities : C1, · · · ,Cn.
Goal : detecting areas where galaxies are ”redder”.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Astronomical data
Observation of a cubic part of the Universe : (20 Mpc)3 .1 Mpc = 3× 1022 metres .
è Locations of the galaxies : X1, · · · ,Xn.
è Light intensities : C1, · · · ,Cn.
Goal : detecting areas where galaxies are ”redder”.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Light intensities
Color
Fre
quen
cy
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
050
100
150
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Light intensities
Color
Fre
quen
cy
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
050
100
150
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Astronomical data
z
x
y
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Astronomical data
z
x
y
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Conclusion
è Graph-based possible clusters, useful in 3D or for multidimensionaldata.
è The MW concentration index is hypothesis-free.
è Simulation study to compare concentration indices.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Conclusion
è Graph-based possible clusters, useful in 3D or for multidimensionaldata.
è The MW concentration index is hypothesis-free.
è Simulation study to compare concentration indices.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Conclusion
è Graph-based possible clusters, useful in 3D or for multidimensionaldata.
è The MW concentration index is hypothesis-free.
è Simulation study to compare concentration indices.
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Conclusion
è Graph-based possible clusters, useful in 3D or for multidimensionaldata.
è The MW concentration index is hypothesis-free.
è Simulation study to compare concentration indices.