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A Mann-Whitney spatial scan statistic for continuous data Lionel - - PowerPoint PPT Presentation

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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results A Mann-Whitney spatial scan statistic for continuous data Lionel Cucala , Christophe Dematte August 24th 2010 Introduction 1-Potential clusters 2-A

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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results

A Mann-Whitney spatial scan statistic for continuous data

Lionel Cucala, Christophe Dematte¨ ı 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

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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

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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results

A data set to analyze

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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 18000 20000 22000 24000 26000 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

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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results

A data set to analyze

➨ 339 dairy farms located in France.

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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results

A data set to analyze

➨ 339 dairy farms located in France. ➨ Somatic individual score (indicator for a disease called mastitis).

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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results

A data set to analyze

➨ 339 dairy farms located in France. ➨ Somatic individual score (indicator for a disease called mastitis). ➨ Marked point process : (Xi, Ci), i = 1, · · · , n where Xi ∈ A ⊂ Rd and Ci ∈ R.

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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

Definition : Cluster= geographical area where the continuous variable is higher than outside.

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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 higher than 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

The null hypothesis

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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results

The null hypothesis

➨ We attempt to reject H0 : (C1, · · · , CN) independent and identically distributed.

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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results

The null hypothesis

➨ We attempt to reject H0 : (C1, · · · , CN) independent and identically distributed. ➨ Goals : detecting cluster(s) and testing the significance according to H0.

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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results

The null hypothesis

➨ We attempt to reject H0 : (C1, · · · , CN) independent and identically distributed. ➨ Goals : detecting cluster(s) and testing the significance according to H0. Two steps :

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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results

The null hypothesis

➨ We attempt to reject H0 : (C1, · · · , CN) independent and identically distributed. ➨ Goals : detecting cluster(s) and testing the significance according to H0. Two steps :

  • defining the set of potential clusters.
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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results

The null hypothesis

➨ We attempt to reject H0 : (C1, · · · , CN) independent and identically distributed. ➨ Goals : detecting cluster(s) and testing the significance according to H0. Two steps :

  • defining the set of potential clusters.
  • choosing a concentration index.
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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results

The null hypothesis

➨ We attempt to reject H0 : (C1, · · · , CN) independent and identically distributed. ➨ Goals : detecting cluster(s) and testing the significance according to H0. Two steps :

  • defining the set of potential clusters.
  • choosing a concentration index.

Statistic : maximal concentration among potential clusters.

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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results

The null hypothesis

➨ We attempt to reject H0 : (C1, · · · , CN) independent and identically distributed. ➨ Goals : detecting cluster(s) and testing the significance according to 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.

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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

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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

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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results

Fixed-shape potential clusters

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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results

Fixed-shape potential clusters

➨ Circles centered on one event, another event on the circumference.

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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results

Fixed-shape potential clusters

➨ Circles centered on one event, another event on the circumference. ➨ Elliptic clusters, with given orientation and shape (2D only).

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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

Graphs G(δ) associated to the point process :

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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 : ➨ Vertices : {1, · · · , 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 : ➨ Vertices : {1, · · · , n}. ➨ 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

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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

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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 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 x y

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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 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 x y

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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 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 x y

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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 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 x y

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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 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 x y

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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 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 x y

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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 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 x y

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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

➨ Connected component of the edge i in G(δ) : Ni(δ).

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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results

Data-based potential clusters

➨ Connected component of the edge i in G(δ) : Ni(δ). ➨ Ai(δ) = {x ∈ A : ∃j ∈ Ni(δ), d(x, xj) ≤ δ}.

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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results

Data-based potential clusters

➨ Connected component of the edge i in G(δ) : Ni(δ). ➨ Ai(δ) = {x ∈ A : ∃j ∈ Ni(δ), d(x, xj) ≤ δ}. ➨ Potential clusters : C= {Ai(δ) : 1 ≤ i ≤ n, δ ∈ R+}.

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Data-based potential clusters

➨ Connected component of the edge i in G(δ) : Ni(δ). ➨ Ai(δ) = {x ∈ A : ∃j ∈ Ni(δ), d(x, xj) ≤ δ}. ➨ Potential clusters : C= {Ai(δ) : 1 ≤ i ≤ n, δ ∈ R+}. ➨ Only n − 1 areas, arbitrarily shaped.

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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

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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

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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results

Concentration indices

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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results

Concentration indices

We evaluate the concentration in Z ⊂ A.

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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results

Concentration indices

We evaluate the concentration in Z ⊂ A. ➨ n(Z) = ♯{i : Xi ∈ Z}.

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Concentration indices

We evaluate the concentration in Z ⊂ A. ➨ n(Z) = ♯{i : Xi ∈ Z}. ➨ µ(Z) =

1 n(Z)

  • i:Xi∈Z Ci.
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Concentration indices

We evaluate the concentration in Z ⊂ A. ➨ n(Z) = ♯{i : Xi ∈ Z}. ➨ µ(Z) =

1 n(Z)

  • i:Xi∈Z Ci.

➨ σ(Z)2 =

1 n(Z)

  • i:Xi∈Z
  • Ci − µ(Z)

2.

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Likelihood-based indices

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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.

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Likelihood-based indices

We test the presence of a cluster in Z ⊂ A. ➨ H0 : Ci ∼ f0.

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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. ➨ H0 : Ci ∼ f0. ➨ H1,Z : Ci|Xi ∼ fZ 1 1(Xi ∈ Z) + f¯

Z

1 1(Xi ∈ ¯ Z).

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Likelihood-based indices

We test the presence of a cluster in Z ⊂ A. ➨ H0 : Ci ∼ f0. ➨ H1,Z : Ci|Xi ∼ fZ 1 1(Xi ∈ Z) + f¯

Z

1 1(Xi ∈ ¯ Z). Likelihood ratio : I(Z) = L1,Z(X1, · · · , Xn, C1, · · · , Cn) L0(X1, · · · , Xn, C1, · · · , Cn) 1 1

  • µ(Z) > µ(¯

Z)

  • .
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Likelihood-based indices

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Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results

Likelihood-based indices

The Exponential model

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Likelihood-based indices

The Exponential model ➨ H0 : Ci ∼ E

  • 1/µ(A)
  • .
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Likelihood-based indices

The Exponential model ➨ H0 : Ci ∼ E

  • 1/µ(A)
  • .

➨ H1,Z : Ci|Xi ∼ E(1/µ(Z)

  • 1

1(Xi ∈ Z) + E

  • 1/µ(¯

Z)

  • 1

1(Xi ∈ ¯ Z).

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Likelihood-based indices

The Exponential model ➨ H0 : Ci ∼ E

  • 1/µ(A)
  • .

➨ H1,Z : Ci|Xi ∼ E(1/µ(Z)

  • 1

1(Xi ∈ Z) + E

  • 1/µ(¯

Z)

  • 1

1(Xi ∈ ¯ Z). Likelihood ratio : Iexp(Z) = −n(Z) log

  • µ(Z)
  • − n(¯

Z) log

  • µ(¯

Z)

  • .
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Likelihood-based indices

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Likelihood-based indices

The homoscedastic Gaussian model

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Likelihood-based indices

The homoscedastic Gaussian model ➨ H0 : Ci ∼ N

  • µ(A), σ(A)2

.

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Likelihood-based indices

The homoscedastic Gaussian model ➨ H0 : Ci ∼ N

  • µ(A), σ(A)2

. ➨ H1,Z : Ci|Xi ∼ N

  • µ(Z), σ2

1,Z

  • 1

1(Xi ∈ Z) +N

  • µ(¯

Z), σ2

1,Z

  • 1

1(Xi ∈ ¯ Z)

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Likelihood-based indices

The homoscedastic Gaussian model ➨ H0 : Ci ∼ N

  • µ(A), σ(A)2

. ➨ H1,Z : Ci|Xi ∼ N

  • µ(Z), σ2

1,Z

  • 1

1(Xi ∈ Z) +N

  • µ(¯

Z), σ2

1,Z

  • 1

1(Xi ∈ ¯ Z) where σ2

1,Z = n(Z)σ(Z)2+n(¯ Z)σ(¯ Z)2 n

.

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Likelihood-based indices

The homoscedastic Gaussian model ➨ H0 : Ci ∼ N

  • µ(A), σ(A)2

. ➨ H1,Z : Ci|Xi ∼ N

  • µ(Z), σ2

1,Z

  • 1

1(Xi ∈ Z) +N

  • µ(¯

Z), σ2

1,Z

  • 1

1(Xi ∈ ¯ Z) where σ2

1,Z = n(Z)σ(Z)2+n(¯ Z)σ(¯ Z)2 n

. Likelihood ratio : Ihomgau(Z) = 1 σ2

1,Z

.

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Likelihood-based indices

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Likelihood-based indices

The heteroscedastic Gaussian model

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Likelihood-based indices

The heteroscedastic Gaussian model ➨ H0 : Ci ∼ N

  • µ(A), σ(A)2

.

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Likelihood-based indices

The heteroscedastic Gaussian model ➨ H0 : Ci ∼ N

  • µ(A), σ(A)2

. ➨ H1,Z : Ci|Xi ∼ N

  • µ(Z), σ(Z)2

1 1(Xi ∈ Z) +N

  • µ(¯

Z), σ(¯ Z)2 1 1(Xi ∈ ¯ Z).

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Likelihood-based indices

The heteroscedastic Gaussian model ➨ H0 : Ci ∼ N

  • µ(A), σ(A)2

. ➨ H1,Z : Ci|Xi ∼ N

  • µ(Z), σ(Z)2

1 1(Xi ∈ Z) +N

  • µ(¯

Z), σ(¯ Z)2 1 1(Xi ∈ ¯ Z). Likelihood ratio : Ihetgau(Z) = −n(Z) log

  • σ(Z)2

− n(¯ Z) log

  • σ(¯

Z)2 .

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No distribution assumption : Mann-Whitney test

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No distribution assumption : Mann-Whitney test ➨ Rj is the rank of Cj among the Ci, 1 ≤ i ≤ n.

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No distribution assumption : Mann-Whitney test ➨ Rj is the rank of Cj among the Ci, 1 ≤ i ≤ n. ➨ RS(Z) =

i:Xi∈Z Ri.

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No distribution assumption : Mann-Whitney test ➨ Rj is the rank of Cj among the Ci, 1 ≤ i ≤ n. ➨ RS(Z) =

i:Xi∈Z Ri.

➨ Under H0, E

  • RS(Z)
  • = M(Z) = n(Z)(n+1)

2

.

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No distribution assumption : Mann-Whitney test ➨ Rj is the rank of Cj among the Ci, 1 ≤ i ≤ n. ➨ RS(Z) =

i:Xi∈Z Ri.

➨ Under H0, E

  • RS(Z)
  • = M(Z) = n(Z)(n+1)

2

. ➨ Under H0, Var

  • RS(Z)
  • = V (Z) = n(Z)n(¯

Z)(n+1) 12

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No distribution assumption : Mann-Whitney test ➨ Rj is the rank of Cj among the Ci, 1 ≤ i ≤ n. ➨ RS(Z) =

i:Xi∈Z Ri.

➨ Under H0, E

  • RS(Z)
  • = M(Z) = n(Z)(n+1)

2

. ➨ Under H0, Var

  • RS(Z)
  • = V (Z) = n(Z)n(¯

Z)(n+1) 12

➨ Under H0, RS(Z)−M(Z) √

V (Z)

− →

d

N(0, 1).

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No distribution assumption : Mann-Whitney test ➨ Rj is the rank of Cj among the Ci, 1 ≤ i ≤ n. ➨ RS(Z) =

i:Xi∈Z Ri.

➨ Under H0, E

  • RS(Z)
  • = M(Z) = n(Z)(n+1)

2

. ➨ Under H0, Var

  • RS(Z)
  • = V (Z) = n(Z)n(¯

Z)(n+1) 12

➨ Under H0, RS(Z)−M(Z) √

V (Z)

− →

d

N(0, 1). Irank(Z) = RS(Z) − M(Z)

  • V (Z)

.

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Outline

Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results

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Outline

Introduction 1-Potential clusters 2-A Mann-Whitney concentration index 3-Results

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The farms data set

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The farms data set

2000 4000 6000 8000 10000 18000 20000 22000 24000 26000 long[1:339] lat[1:339]

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The farms data set

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The farms data set

2000 4000 6000 8000 10000 18000 20000 22000 24000 26000 long[1:339] lat[1:339]

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Astronomical data

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Astronomical data

Observation of a cubic part of the Universe : (20 Mpc)3 .

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Astronomical data

Observation of a cubic part of the Universe : (20 Mpc)3 . 1 Mpc = 3 × 1022 metres .

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Astronomical data

Observation of a cubic part of the Universe : (20 Mpc)3 . 1 Mpc = 3 × 1022 metres . ➨ Locations of the galaxies : X1, · · · , Xn.

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Astronomical data

Observation of a cubic part of the Universe : (20 Mpc)3 . 1 Mpc = 3 × 1022 metres . ➨ Locations of the galaxies : X1, · · · , Xn. ➨ Light intensities : C1, · · · , Cn.

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Astronomical data

Observation of a cubic part of the Universe : (20 Mpc)3 . 1 Mpc = 3 × 1022 metres . ➨ Locations of the galaxies : X1, · · · , Xn. ➨ Light intensities : C1, · · · , Cn. Goal : detecting areas where galaxies are ”redder”.

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Light intensities

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Light intensities

Color Frequency 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 50 100 150

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Astronomical data

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Astronomical data

z x y

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Conclusion

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Conclusion

➨ Graph-based possible clusters, useful in 3D or for multidimensional data.

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Conclusion

➨ Graph-based possible clusters, useful in 3D or for multidimensional data. ➨ The MW concentration index is hypothesis-free.

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Conclusion

➨ Graph-based possible clusters, useful in 3D or for multidimensional data. ➨ The MW concentration index is hypothesis-free. ➨ Simulation study to compare concentration indices.