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

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

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

−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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.