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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
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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
![Page 3: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/3.jpg)
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
![Page 4: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/4.jpg)
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]
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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]
![Page 6: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/6.jpg)
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.
![Page 7: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/7.jpg)
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.
![Page 8: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/8.jpg)
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.
![Page 9: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/9.jpg)
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.
![Page 10: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/10.jpg)
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 ?
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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 ?
![Page 12: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/12.jpg)
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 ?
![Page 13: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/13.jpg)
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.
![Page 14: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/14.jpg)
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.
![Page 15: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/15.jpg)
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.
![Page 16: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/16.jpg)
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.
![Page 17: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/17.jpg)
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.
![Page 18: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/18.jpg)
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.
![Page 19: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/19.jpg)
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.
![Page 20: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/20.jpg)
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.
![Page 21: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/21.jpg)
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
![Page 22: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/22.jpg)
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
![Page 23: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/23.jpg)
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).
![Page 24: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/24.jpg)
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).
![Page 25: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/25.jpg)
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).
![Page 26: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/26.jpg)
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}.
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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}.
![Page 28: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/28.jpg)
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}.
![Page 29: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/29.jpg)
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}.
![Page 30: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/30.jpg)
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
−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
−0.5 0.0 0.5 1.0 1.5
−1.
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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
−1.
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0.5
0.0
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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
−1.
0−
0.5
0.0
0.5
1.0
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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
−1.
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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
−1.
0−
0.5
0.0
0.5
1.0
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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
−1.
0−
0.5
0.0
0.5
1.0
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![Page 40: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/40.jpg)
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
−1.
0−
0.5
0.0
0.5
1.0
1.5
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![Page 41: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/41.jpg)
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
−1.
0−
0.5
0.0
0.5
1.0
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y
![Page 42: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/42.jpg)
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
−1.
0−
0.5
0.0
0.5
1.0
1.5
2.0
x
y
![Page 43: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/43.jpg)
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
−1.
0−
0.5
0.0
0.5
1.0
1.5
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y
![Page 44: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/44.jpg)
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
−1.
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0.0
0.5
1.0
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y
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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
−1.
0−
0.5
0.0
0.5
1.0
1.5
2.0
x
y
![Page 46: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/46.jpg)
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.
![Page 47: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/47.jpg)
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.
![Page 48: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/48.jpg)
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.
![Page 49: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/49.jpg)
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.
![Page 50: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/50.jpg)
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.
![Page 51: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/51.jpg)
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
![Page 52: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/52.jpg)
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
![Page 53: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/53.jpg)
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.
![Page 54: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/54.jpg)
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.
![Page 55: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/55.jpg)
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.
![Page 56: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/56.jpg)
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.
![Page 57: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/57.jpg)
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.
![Page 58: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/58.jpg)
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 )
).
![Page 59: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/59.jpg)
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 )
).
![Page 60: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/60.jpg)
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 )
).
![Page 61: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/61.jpg)
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 )
).
![Page 62: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/62.jpg)
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 )
).
![Page 63: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/63.jpg)
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 )
).
![Page 64: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/64.jpg)
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 )
).
![Page 65: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/65.jpg)
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 )
).
![Page 66: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/66.jpg)
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 )
).
![Page 67: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/67.jpg)
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 )
).
![Page 68: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/68.jpg)
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
.
![Page 69: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/69.jpg)
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
.
![Page 70: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/70.jpg)
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
.
![Page 71: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/71.jpg)
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
.
![Page 72: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/72.jpg)
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
.
![Page 73: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/73.jpg)
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
.
![Page 74: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/74.jpg)
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
).
![Page 75: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/75.jpg)
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
).
![Page 76: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/76.jpg)
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
).
![Page 77: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/77.jpg)
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
).
![Page 78: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/78.jpg)
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
).
![Page 79: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/79.jpg)
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 ).
![Page 80: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/80.jpg)
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 ).
![Page 81: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/81.jpg)
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 ).
![Page 82: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/82.jpg)
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 ).
![Page 83: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/83.jpg)
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 ).
![Page 84: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/84.jpg)
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 ).
![Page 85: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/85.jpg)
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 ).
![Page 86: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/86.jpg)
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 ).
![Page 87: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/87.jpg)
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
![Page 88: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/88.jpg)
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
![Page 89: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/89.jpg)
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]
![Page 90: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/90.jpg)
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]
![Page 91: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/91.jpg)
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]
![Page 92: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/92.jpg)
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]
![Page 93: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/93.jpg)
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”.
![Page 94: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/94.jpg)
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”.
![Page 95: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/95.jpg)
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”.
![Page 96: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/96.jpg)
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”.
![Page 97: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/97.jpg)
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”.
![Page 98: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/98.jpg)
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”.
![Page 99: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/99.jpg)
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
![Page 100: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/100.jpg)
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
![Page 101: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/101.jpg)
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Astronomical data
z
x
y
![Page 102: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/102.jpg)
Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results
Astronomical data
z
x
y
![Page 103: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/103.jpg)
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.
![Page 104: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/104.jpg)
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.
![Page 105: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/105.jpg)
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.
![Page 106: A Mann-Whitney spatial scan statistic for continuous data · 2010. 8. 4. · 2.0 x y. Introduction1-Potential clusters2-A Mann-Whitney concentration index3-Results Data-based potential](https://reader035.vdocuments.net/reader035/viewer/2022071508/6129378a07d0627e881e51d4/html5/thumbnails/106.jpg)
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.