A general notion of interiority 6th Winter S hool on Te hnology Assessment � 1

A general notion of interiority

J. Orestes Cerdeira

Fa uldade de Ciên ias e Te nologia,

Universidade Nova de Lisboa,

Portugal

Joint with M. João Martins, Pedro C. Silva and T. Monteiro-Henriques

two issues...

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 2

to what extent a set of entities is separated from ?

how are the entities of a group arranged together ?

(more on entrated to the "interior� or dispersed on the

"margins� of their range)

two issues...

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 2

� to what extent a set of entities {•} is separated from • ?

how are the entities of a group arranged together ?

(more on entrated to the "interior� or dispersed on the

"margins� of their range)

two issues...

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 2

� to what extent a set of entities {•} is separated from • ?

how are the entities of a group arranged together ?

(more on entrated to the "interior� or dispersed on the

"margins� of their range)

two issues...

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 2

� to what extent a set of entities {•} is separated from • ?

how are the entities of a group arranged together ?

(more on entrated to the "interior� or dispersed on the

"margins� of their range)

two issues...

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 2

� to what extent a set of entities {•} is separated from • ?

� how are the entities of a group arranged together ?

(more on entrated to the "interior� or dispersed on the

"margins� of their range)

two issues...

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 2

� to what extent a set of entities {•} is separated from • ?

� how are the entities of a group arranged together ?

(more on entrated to the "interior� or dispersed on the

"margins� of their range)

two issues...

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 2

� to what extent a set of entities {•} is separated from • ?

� how are the entities of a group arranged together ?

(more on entrated to the "interior� or dispersed on the

"margins� of their range)

two issues...

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 2

� to what extent a set of entities {•} is separated from • ?

� how are the entities of a group arranged together ?

(more on entrated to the "interior� or dispersed on the

"margins� of their range)

issue 1: is {•} well separated from {•}?

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 3

issue 1: is {•} well separated from {•}?

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 3

In many areas ...

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 4

E ology, Chemistry, Mole ular Biology, Physiology, Toxi ology,

So ial S ien es ...

issue: to what extent a parti ular subset of entities is

separated from the other entities? arises

In many areas ...

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 4

E ology, Chemistry, Mole ular Biology, Physiology, Toxi ology,

So ial S ien es ...

issue: to what extent a parti ular subset of entities is

separated from the other entities? arises

In many areas ...

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 4

E ology, Chemistry, Mole ular Biology, Physiology, Toxi ology,

So ial S ien es ...

issue:

to what extent a parti ular subset of entities is

separated from the other entities? arises

In many areas ...

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 4

E ology, Chemistry, Mole ular Biology, Physiology, Toxi ology,

So ial S ien es ...

issue: to what extent a parti ular subset X of entities is

separated from the other entities?

arises

In many areas ...

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 4

E ology, Chemistry, Mole ular Biology, Physiology, Toxi ology,

So ial S ien es ...

issue: to what extent a parti ular subset X of entities is

separated from the other entities? arises

In the ontext of luster analysis...

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 5

methods have been proposed that measure overlap/separability of

lusters as part of assessing the quality of a given partition.

Dunn's index and Dunn-like indi es, Davies-Bouldin's index, the

silhouette index and Thornton's index.

With these approa hes separability is numeri ally quanti�ed, but

no detailed information on how ea h luster is separated from

the others is produ ed

all, ex ept Thornton's index, are expressions that expli itly

involve numeri al dissimilarities between entities

notions su h as �perfe t separation of luster� are not

omprise.

In the ontext of luster analysis...

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 5

methods have been proposed that measure overlap/separability of

lusters as part of assessing the quality of a given partition.

Dunn's index and Dunn-like indi es, Davies-Bouldin's index, the

silhouette index and Thornton's index.

With these approa hes separability is numeri ally quanti�ed, but

no detailed information on how ea h luster is separated from

the others is produ ed

all, ex ept Thornton's index, are expressions that expli itly

involve numeri al dissimilarities between entities

notions su h as �perfe t separation of luster� are not

omprise.

In the ontext of luster analysis...

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 5

methods have been proposed that measure overlap/separability of

lusters as part of assessing the quality of a given partition.

Dunn's index and Dunn-like indi es, Davies-Bouldin's index, the

silhouette index and Thornton's index.

With these approa hes separability is numeri ally quanti�ed

, but

no detailed information on how ea h luster is separated from

the others is produ ed

all, ex ept Thornton's index, are expressions that expli itly

involve numeri al dissimilarities between entities

notions su h as �perfe t separation of luster� are not

omprise.

In the ontext of luster analysis...

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 5

methods have been proposed that measure overlap/separability of

lusters as part of assessing the quality of a given partition.

Dunn's index and Dunn-like indi es, Davies-Bouldin's index, the

silhouette index and Thornton's index.

With these approa hes separability is numeri ally quanti�ed, but

no detailed information on how ea h luster is separated from

the others is produ ed

all, ex ept Thornton's index, are expressions that expli itly

involve numeri al dissimilarities between entities

notions su h as �perfe t separation of luster� are not

omprise.

In the ontext of luster analysis...

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 5

methods have been proposed that measure overlap/separability of

lusters as part of assessing the quality of a given partition.

Dunn's index and Dunn-like indi es, Davies-Bouldin's index, the

silhouette index and Thornton's index.

With these approa hes separability is numeri ally quanti�ed, but

� no detailed information on how ea h luster is separated from

the others is produ ed

all, ex ept Thornton's index, are expressions that expli itly

involve numeri al dissimilarities between entities

notions su h as �perfe t separation of luster� are not

omprise.

In the ontext of luster analysis...

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 5

methods have been proposed that measure overlap/separability of

lusters as part of assessing the quality of a given partition.

Dunn's index and Dunn-like indi es, Davies-Bouldin's index, the

silhouette index and Thornton's index.

With these approa hes separability is numeri ally quanti�ed, but

� no detailed information on how ea h luster is separated from

the others is produ ed

� all, ex ept Thornton's index, are expressions that expli itly

involve numeri al dissimilarities between entities

notions su h as �perfe t separation of luster� are not

omprise.

In the ontext of luster analysis...

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 5

methods have been proposed that measure overlap/separability of

lusters as part of assessing the quality of a given partition.

Dunn's index and Dunn-like indi es, Davies-Bouldin's index, the

silhouette index and Thornton's index.

With these approa hes separability is numeri ally quanti�ed, but

� no detailed information on how ea h luster is separated from

the others is produ ed

� all, ex ept Thornton's index, are expressions that expli itly

involve numeri al dissimilarities between entities

� notions su h as �perfe t separation of luster� are not

omprise.

Silhouette index

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 6

To onstru t the silhouette S(X) (with respe t to X̄), for i ∈ X

S(i) =b(i)− a(i)

max{a(i), b(i)}

a(i) - average dissimilarity of entity i to all other entities in X

b(i) - average dissimilarity of entity i to all entities in X̄

.

, means is �well- lustered� in

, means lies �equally far away� from and

, means is �mis lassi�ed� as an entity.

is the average of the silhouettes of

notions su h as �perfe t separation of luster� are not omprise.

Silhouette index

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 6

To onstru t the silhouette S(X) (with respe t to X̄), for i ∈ X

S(i) =b(i)− a(i)

max{a(i), b(i)}

a(i) - average dissimilarity of entity i to all other entities in X

b(i) - average dissimilarity of entity i to all entities in X̄

S(i) ∈ [−1, 1].

, means is �well- lustered� in

, means lies �equally far away� from and

, means is �mis lassi�ed� as an entity.

is the average of the silhouettes of

notions su h as �perfe t separation of luster� are not omprise.

Silhouette index

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 6

To onstru t the silhouette S(X) (with respe t to X̄), for i ∈ X

S(i) =b(i)− a(i)

max{a(i), b(i)}

a(i) - average dissimilarity of entity i to all other entities in X

b(i) - average dissimilarity of entity i to all entities in X̄

S(i) ∈ [−1, 1].

S(i) ≃ 1, means i is �well- lustered� in X

, means lies �equally far away� from and

, means is �mis lassi�ed� as an entity.

is the average of the silhouettes of

notions su h as �perfe t separation of luster� are not omprise.

Silhouette index

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 6

To onstru t the silhouette S(X) (with respe t to X̄), for i ∈ X

S(i) =b(i)− a(i)

max{a(i), b(i)}

a(i) - average dissimilarity of entity i to all other entities in X

b(i) - average dissimilarity of entity i to all entities in X̄

S(i) ∈ [−1, 1].

S(i) ≃ 1, means i is �well- lustered� in X

S(i) ≃ 0, means i lies �equally far away� from X and X̄

, means is �mis lassi�ed� as an entity.

is the average of the silhouettes of

notions su h as �perfe t separation of luster� are not omprise.

Silhouette index

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 6

To onstru t the silhouette S(X) (with respe t to X̄), for i ∈ X

S(i) =b(i)− a(i)

max{a(i), b(i)}

a(i) - average dissimilarity of entity i to all other entities in X

b(i) - average dissimilarity of entity i to all entities in X̄

S(i) ∈ [−1, 1].

S(i) ≃ 1, means i is �well- lustered� in X

S(i) ≃ 0, means i lies �equally far away� from X and X̄

S(i) ≃ −1, means i is �mis lassi�ed� as an X entity.

is the average of the silhouettes of

notions su h as �perfe t separation of luster� are not omprise.

Silhouette index

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 6

To onstru t the silhouette S(X) (with respe t to X̄), for i ∈ X

S(i) =b(i)− a(i)

max{a(i), b(i)}

a(i) - average dissimilarity of entity i to all other entities in X

b(i) - average dissimilarity of entity i to all entities in X̄

S(i) ∈ [−1, 1].

S(i) ≃ 1, means i is �well- lustered� in X

S(i) ≃ 0, means i lies �equally far away� from X and X̄

S(i) ≃ −1, means i is �mis lassi�ed� as an X entity.

S(X) is the average of the silhouettes of i ∈ X

notions su h as �perfe t separation of luster� are not omprise.

Silhouette index

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 6

To onstru t the silhouette S(X) (with respe t to X̄), for i ∈ X

S(i) =b(i)− a(i)

max{a(i), b(i)}

a(i) - average dissimilarity of entity i to all other entities in X

b(i) - average dissimilarity of entity i to all entities in X̄

S(i) ∈ [−1, 1].

S(i) ≃ 1, means i is �well- lustered� in X

S(i) ≃ 0, means i lies �equally far away� from X and X̄

S(i) ≃ −1, means i is �mis lassi�ed� as an X entity.

S(X) is the average of the silhouettes of i ∈ X

notions su h as �perfe t separation of luster� are not omprise.

Thornton's index

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 7

if the nearest neighbor of is (not) in

Thornton's index

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 7

GSI =

∑i∈X δi

|X|, δi = 1(0) if the nearest neighbor of i is (not) in X

Thornton's index

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 7

GSI =

∑i∈X δi

|X|, δi = 1(0) if the nearest neighbor of i is (not) in X

Thornton's index

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 7

GSI =

∑i∈X δi

|X|, δi = 1(0) if the nearest neighbor of i is (not) in X

GSI = 1

Thornton's index

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 7

GSI =

∑i∈X δi

|X|, δi = 1(0) if the nearest neighbor of i is (not) in X

GSI = 1

Thornton's index

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 7

GSI =

∑i∈X δi

|X|, δi = 1(0) if the nearest neighbor of i is (not) in X

GSI = 1

GSI = 1

other proposals rely on a notion of territory

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 8

doesn't dis riminate separability

other proposals rely on a notion of territory

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 8

doesn't dis riminate separability

other proposals rely on a notion of territory

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 8

doesn't dis riminate separability

other proposals rely on a notion of territory

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 8

doesn't dis riminate separability

Instead of deeming...

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 9

to what extent a set of entities X is separated from other

entities?

to what extent a set of entities is separated from another

entity?

Instead of deeming...

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 9

to what extent a set of entities X is separated from other

entities?

to what extent a set of entities X is separated from another

entity?

to what extend {•} is separated from • ?

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 10

to what extend {•} is separated from • ?

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 10

to what extend {•} is separated from • ?

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 10

to what extend {•} is separated from • ?

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 10

to what extend {•} is separated from • ?

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 10

to formalize interiority needs �xing some aggregation riterion

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 11

i.e., a rule that identi�es, for a set of entities and ,

the -partitions of whi h are adequately aggregated

is adequate

if the sum of distan es from ea h point to its luster enter is

minimum ( -means)

if the largest diameter of its lusters is minimum (diameter)

if lusters are the onne ted omponents of some minimum

-spanning forest (single linkage)

to formalize interiority needs �xing some aggregation riterion

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 11

i.e., a rule that identi�es, for a set of entities E and 1 ≤ k ≤ |E|,

the k-partitions of E whi h are adequately aggregated

is adequate

if the sum of distan es from ea h point to its luster enter is

minimum ( -means)

if the largest diameter of its lusters is minimum (diameter)

if lusters are the onne ted omponents of some minimum

-spanning forest (single linkage)

to formalize interiority needs �xing some aggregation riterion

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 11

i.e., a rule that identi�es, for a set of entities E and 1 ≤ k ≤ |E|,

the k-partitions of E whi h are adequately aggregated

Pk is adequate

if the sum of distan es from ea h point to its luster enter is

minimum ( -means)

if the largest diameter of its lusters is minimum (diameter)

if lusters are the onne ted omponents of some minimum

-spanning forest (single linkage)

to formalize interiority needs �xing some aggregation riterion

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 11

i.e., a rule that identi�es, for a set of entities E and 1 ≤ k ≤ |E|,

the k-partitions of E whi h are adequately aggregated

Pk is adequate

� if the sum of distan es from ea h point to its luster enter is

minimum

( -means)

if the largest diameter of its lusters is minimum (diameter)

if lusters are the onne ted omponents of some minimum

-spanning forest (single linkage)

to formalize interiority needs �xing some aggregation riterion

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 11

i.e., a rule that identi�es, for a set of entities E and 1 ≤ k ≤ |E|,

the k-partitions of E whi h are adequately aggregated

Pk is adequate

� if the sum of distan es from ea h point to its luster enter is

minimum (k-means)

if the largest diameter of its lusters is minimum (diameter)

if lusters are the onne ted omponents of some minimum

-spanning forest (single linkage)

to formalize interiority needs �xing some aggregation riterion

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 11

i.e., a rule that identi�es, for a set of entities E and 1 ≤ k ≤ |E|,

the k-partitions of E whi h are adequately aggregated

Pk is adequate

� if the sum of distan es from ea h point to its luster enter is

minimum (k-means)

� if the largest diameter of its lusters is minimum

(diameter)

if lusters are the onne ted omponents of some minimum

-spanning forest (single linkage)

to formalize interiority needs �xing some aggregation riterion

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 11

i.e., a rule that identi�es, for a set of entities E and 1 ≤ k ≤ |E|,

the k-partitions of E whi h are adequately aggregated

Pk is adequate

� if the sum of distan es from ea h point to its luster enter is

minimum (k-means)

� if the largest diameter of its lusters is minimum (diameter)

if lusters are the onne ted omponents of some minimum

-spanning forest (single linkage)

to formalize interiority needs �xing some aggregation riterion

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 11

i.e., a rule that identi�es, for a set of entities E and 1 ≤ k ≤ |E|,

the k-partitions of E whi h are adequately aggregated

Pk is adequate

� if the sum of distan es from ea h point to its luster enter is

minimum (k-means)

� if the largest diameter of its lusters is minimum (diameter)

� if lusters are the onne ted omponents of some minimum

(|E| − k)-spanning forest

(single linkage)

to formalize interiority needs �xing some aggregation riterion

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 11

i.e., a rule that identi�es, for a set of entities E and 1 ≤ k ≤ |E|,

the k-partitions of E whi h are adequately aggregated

Pk is adequate

� if the sum of distan es from ea h point to its luster enter is

minimum (k-means)

� if the largest diameter of its lusters is minimum (diameter)

� if lusters are the onne ted omponents of some minimum

(|E| − k)-spanning forest (single linkage)

to formalize interiority needs �xing some aggregation riterion

⊲ Introdu tion

Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 11

i.e., a rule that identi�es, for a set of entities E and 1 ≤ k ≤ |E|,

the k-partitions of E whi h are adequately aggregated

Pk is adequate

� if the sum of distan es from ea h point to its luster enter is

minimum (k-means)

� if the largest diameter of its lusters is minimum (diameter)

� if lusters are the onne ted omponents of some minimum

(|E| − k)-spanning forest (single linkage)

� · · ·

k-interior

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 12

entity x̄ is k-interior with respe t to X if for every k-partition Pk

of X, Pk ∪{{x̄}} is not an adequate (k+1)-partition of X ∪{x̄}

is 1-interior

is 2-interior

k-interior

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 12

entity x̄ is k-interior with respe t to X if for every k-partition Pk

of X, Pk ∪{{x̄}} is not an adequate (k+1)-partition of X ∪{x̄}

is

1-interior

is 2-interior

k-interior

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 12

entity x̄ is k-interior with respe t to X if for every k-partition Pk

of X, Pk ∪{{x̄}} is not an adequate (k+1)-partition of X ∪{x̄}

is

1-interior

is 2-interior

k-interior

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 12

entity x̄ is k-interior with respe t to X if for every k-partition Pk

of X, Pk ∪{{x̄}} is not an adequate (k+1)-partition of X ∪{x̄}

is 1-interior

is 2-interior

k-interior

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 12

entity x̄ is k-interior with respe t to X if for every k-partition Pk

of X, Pk ∪{{x̄}} is not an adequate (k+1)-partition of X ∪{x̄}

is

1-interior

is 2-interior

k-interior

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 12

entity x̄ is k-interior with respe t to X if for every k-partition Pk

of X, Pk ∪{{x̄}} is not an adequate (k+1)-partition of X ∪{x̄}

is 1-interior

is 2-interior

k-interior

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 12

entity x̄ is k-interior with respe t to X if for every k-partition Pk

of X, Pk ∪{{x̄}} is not an adequate (k+1)-partition of X ∪{x̄}

• is 1-interior

is 2-interior

k-interior

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 12

entity x̄ is k-interior with respe t to X if for every k-partition Pk

of X, Pk ∪{{x̄}} is not an adequate (k+1)-partition of X ∪{x̄}

• is 1-interior

is 2-interior

k-interior

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 12

entity x̄ is k-interior with respe t to X if for every k-partition Pk

of X, Pk ∪{{x̄}} is not an adequate (k+1)-partition of X ∪{x̄}

• is 1-interior

is 2-interior

k-interior

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 12

entity x̄ is k-interior with respe t to X if for every k-partition Pk

of X, Pk ∪{{x̄}} is not an adequate (k+1)-partition of X ∪{x̄}

• is 1-interior

is 2-interior

k-interior

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 12

entity x̄ is k-interior with respe t to X if for every k-partition Pk

of X, Pk ∪{{x̄}} is not an adequate (k+1)-partition of X ∪{x̄}

• is 1-interior

is 2-interior

k-interior

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 12

entity x̄ is k-interior with respe t to X if for every k-partition Pk

of X, Pk ∪{{x̄}} is not an adequate (k+1)-partition of X ∪{x̄}

• is 1-interior

is 2-interior

k-interior

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 12

entity x̄ is k-interior with respe t to X if for every k-partition Pk

of X, Pk ∪{{x̄}} is not an adequate (k+1)-partition of X ∪{x̄}

• is 1-interior

• is 2-interior

k-interior

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 13

entity x̄ is k-interior with respe t to X if for every k-partition Pk

of X, Pk ∪{{x̄}} is not an adequate (k+1)-partition of X ∪{x̄}

is 1-interior but not 2-interior

k-interior

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 13

entity x̄ is k-interior with respe t to X if for every k-partition Pk

of X, Pk ∪{{x̄}} is not an adequate (k+1)-partition of X ∪{x̄}

is 1-interior but not 2-interior

k-interior

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 13

entity x̄ is k-interior with respe t to X if for every k-partition Pk

of X, Pk ∪{{x̄}} is not an adequate (k+1)-partition of X ∪{x̄}

is 1-interior but not 2-interior

k-interior

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 13

entity x̄ is k-interior with respe t to X if for every k-partition Pk

of X, Pk ∪{{x̄}} is not an adequate (k+1)-partition of X ∪{x̄}

• is 1-interior

but not 2-interior

k-interior

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 13

entity x̄ is k-interior with respe t to X if for every k-partition Pk

of X, Pk ∪{{x̄}} is not an adequate (k+1)-partition of X ∪{x̄}

• is 1-interior

but not 2-interior

k-interior

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 13

entity x̄ is k-interior with respe t to X if for every k-partition Pk

of X, Pk ∪{{x̄}} is not an adequate (k+1)-partition of X ∪{x̄}

• is 1-interior but not 2-interior

interiority degree

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 14

the interiority degree of is

the largest su h that is -interior

if is not interior

� diameter i�

� single linkage i�

,

where is a non trivial bipartition of

diameter and single linkage i�

interiority degree

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 14

the interiority degree of x̄ is

dx̄ = the largest k su h that x̄ is k-interior

if is not interior

� diameter i�

� single linkage i�

,

where is a non trivial bipartition of

diameter and single linkage i�

interiority degree

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 14

the interiority degree of x̄ is

dx̄ = the largest k su h that x̄ is k-interior

dx̄ = 0 if x̄ is not interior

� diameter i�

� single linkage i�

,

where is a non trivial bipartition of

diameter and single linkage i�

interiority degree

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 14

the interiority degree of x̄ is

dx̄ = the largest k su h that x̄ is k-interior

dx̄ = 0 if x̄ is not interior

� dx̄ = 0

� diameter i�

� single linkage i�

,

where is a non trivial bipartition of

diameter and single linkage i�

interiority degree

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 14

the interiority degree of x̄ is

dx̄ = the largest k su h that x̄ is k-interior

dx̄ = 0 if x̄ is not interior

� dx̄ = 0

� diameter i�

� single linkage i�

,

where is a non trivial bipartition of

diameter and single linkage i�

interiority degree

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 14

the interiority degree of x̄ is

dx̄ = the largest k su h that x̄ is k-interior

dx̄ = 0 if x̄ is not interior

� dx̄ = 0

� diameter i� minx∈X

dis(x̄, x) ≥ maxx,x′∈X

dis(x, x′)

� single linkage i�

,

where is a non trivial bipartition of

diameter and single linkage i�

interiority degree

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 14

the interiority degree of x̄ is

dx̄ = the largest k su h that x̄ is k-interior

dx̄ = 0 if x̄ is not interior

� dx̄ = 0

� diameter i� minx∈X

dis(x̄, x) ≥ maxx,x′∈X

dis(x, x′)

� single linkage i�

,

where is a non trivial bipartition of

diameter and single linkage i�

interiority degree

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 14

the interiority degree of x̄ is

dx̄ = the largest k su h that x̄ is k-interior

dx̄ = 0 if x̄ is not interior

� dx̄ = 0

� diameter i� minx∈X

dis(x̄, x) ≥ maxx,x′∈X

dis(x, x′)

� single linkage i�

minx∈X

dis(x̄, x) ≥ max{X′,X′′}

minx′∈X′,x′′∈X′′

dis(x′, x′′),

where {X ′, X ′′} is a non trivial bipartition of X

diameter and single linkage i�

interiority degree

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 14

the interiority degree of x̄ is

dx̄ = the largest k su h that x̄ is k-interior

dx̄ = 0 if x̄ is not interior

� dx̄ = 0

� diameter i� minx∈X

dis(x̄, x) ≥ maxx,x′∈X

dis(x, x′)

� single linkage i�

minx∈X

dis(x̄, x) ≥ max{X′,X′′}

minx′∈X′,x′′∈X′′

dis(x′, x′′),

where {X ′, X ′′} is a non trivial bipartition of X

� dx̄ = |X| − 1 diameter and single linkage i�

interiority degree

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 14

the interiority degree of x̄ is

dx̄ = the largest k su h that x̄ is k-interior

dx̄ = 0 if x̄ is not interior

� dx̄ = 0

� diameter i� minx∈X

dis(x̄, x) ≥ maxx,x′∈X

dis(x, x′)

� single linkage i�

minx∈X

dis(x̄, x) ≥ max{X′,X′′}

minx′∈X′,x′′∈X′′

dis(x′, x′′),

where {X ′, X ′′} is a non trivial bipartition of X

� dx̄ = |X| − 1 diameter and single linkage i�

minx∈X

dis(x̄, x) < minx,x′∈X

dis(x, x′)

this ombinatorial on ept of interiority...

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 15

de�nes, for ea h aggregation riterion, a parti ular partition of

the entity spa e into a �nite number of iso-interiority regions

this ombinatorial on ept of interiority...

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 15

de�nes, for ea h aggregation riterion, a parti ular partition of

the entity spa e into a �nite number of iso-interiority regions

this ombinatorial on ept of interiority...

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 15

de�nes, for ea h aggregation riterion, a parti ular partition of

the entity spa e into a �nite number of iso-interiority regions

d : E \X → {0, 1, . . . , |X| − 1}

this ombinatorial on ept of interiority...

Introdu tion

⊲ Interiority

Separability index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 15

de�nes, for ea h aggregation riterion, a parti ular partition of

the entity spa e into a �nite number of iso-interiority regions

d : E \X → {0, 1, . . . , |X| − 1}

separability index

Introdu tion

Interiority

⊲

Separability

index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 16

interiority of

dis riminate sep. index bla k/(bla k+red)

separability index

Introdu tion

Interiority

⊲

Separability

index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 16

DX̄ = {d(x̄) : x̄ ∈ X̄ = {•}} interiority of X̄

dis riminate sep. index bla k/(bla k+red)

separability index

Introdu tion

Interiority

⊲

Separability

index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 16

DX̄ = {d(x̄) : x̄ ∈ X̄ = {•}} interiority of X̄

DX̄ dis riminate sep. index DSI ∈ [0, 1]

dis riminate sep. index bla k/(bla k+red)

separability index

Introdu tion

Interiority

⊲

Separability

index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 16

DX̄ = {d(x̄) : x̄ ∈ X̄ = {•}} interiority of X̄

DX̄ dis riminate sep. index DSI ∈ [0, 1]

DSI of m points with interiority degree n = DSI of n points

with interiority degree m

dis riminate sep. index bla k/(bla k+red)

separability index

Introdu tion

Interiority

⊲

Separability

index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 16

DX̄ = {d(x̄) : x̄ ∈ X̄ = {•}} interiority of X̄

DX̄ dis riminate sep. index DSI ∈ [0, 1]

dis riminate sep. index bla k/(bla k+red)

separability index

Introdu tion

Interiority

⊲

Separability

index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 16

DX̄ = {d(x̄) : x̄ ∈ X̄ = {•}} interiority of X̄

DX̄ dis riminate sep. index DSI ∈ [0, 1]

dis riminate sep. index bla k/(bla k+red)

separability index

Introdu tion

Interiority

⊲

Separability

index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 16

DX̄ = {d(x̄) : x̄ ∈ X̄ = {•}} interiority of X̄

DX̄ dis riminate sep. index DSI = bla k/(bla k+red)

DSI = bla k/(bla k+red)

Introdu tion

Interiority

⊲

Separability

index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 17

DSI = bla k/(bla k+red)

Introdu tion

Interiority

⊲

Separability

index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 17

DSI = bla k/(bla k+red)

Introdu tion

Interiority

⊲

Separability

index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 17

DSI = bla k/(bla k+red)

Introdu tion

Interiority

⊲

Separability

index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 18

DSI = bla k/(bla k+red)

Introdu tion

Interiority

⊲

Separability

index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 18

DSI = bla k/(bla k+red)

Introdu tion

Interiority

⊲

Separability

index

Convexity riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 18

onvexity riterion

Introdu tion

Interiority

Separability index

⊲

Convexity

riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 19

a -partition is adequate if it in ludes singletons that are

not in the interior of the onvex hull of the remaining points

onvexity riterion

Introdu tion

Interiority

Separability index

⊲

Convexity

riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 19

a k-partition is adequate if it in ludes k − 1 singletons that are

not in the interior of the onvex hull of the remaining points

onvexity riterion

Introdu tion

Interiority

Separability index

⊲

Convexity

riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 19

a k-partition is adequate if it in ludes k − 1 singletons that are

not in the interior of the onvex hull of the remaining points

onvexity riterion

Introdu tion

Interiority

Separability index

⊲

Convexity

riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 19

a k-partition is adequate if it in ludes k − 1 singletons that are

not in the interior of the onvex hull of the remaining points

onvexity riterion

Introdu tion

Interiority

Separability index

⊲

Convexity

riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 19

a k-partition is adequate if it in ludes k − 1 singletons that are

not in the interior of the onvex hull of the remaining points

not adequate

onvexity riterion

Introdu tion

Interiority

Separability index

⊲

Convexity

riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 19

a k-partition is adequate if it in ludes k − 1 singletons that are

not in the interior of the onvex hull of the remaining points

not adequate

onvexity riterion

Introdu tion

Interiority

Separability index

⊲

Convexity

riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 19

a k-partition is adequate if it in ludes k − 1 singletons that are

not in the interior of the onvex hull of the remaining points

onvexity riterion

Introdu tion

Interiority

Separability index

⊲

Convexity

riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 19

a k-partition is adequate if it in ludes k − 1 singletons that are

not in the interior of the onvex hull of the remaining points

adequate

onvexity riterion

Introdu tion

Interiority

Separability index

⊲

Convexity

riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 19

a k-partition is adequate if it in ludes k − 1 singletons that are

not in the interior of the onvex hull of the remaining points

adequate

onvexity riterion

Introdu tion

Interiority

Separability index

⊲

Convexity

riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 19

a k-partition is adequate if it in ludes k − 1 singletons that are

not in the interior of the onvex hull of the remaining points

onvexity riterion

Introdu tion

Interiority

Separability index

⊲

Convexity

riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 19

a k-partition is adequate if it in ludes k − 1 singletons that are

not in the interior of the onvex hull of the remaining points

adequate

onvexity riterion

Introdu tion

Interiority

Separability index

⊲

Convexity

riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 19

a k-partition is adequate if it in ludes k − 1 singletons that are

not in the interior of the onvex hull of the remaining points

adequate

onvexity riterion

Introdu tion

Interiority

Separability index

⊲

Convexity

riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 20

x̄ is k-interior if it is inside the onvex hull of any |X| − k + 1

points of X

- (halfspa e, lo ation, Tukey) depth

onvexity riterion

Introdu tion

Interiority

Separability index

⊲

Convexity

riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 20

x̄ is k-interior if it is inside the onvex hull of any |X| − k + 1

points of X

- (halfspa e, lo ation, Tukey) depth

onvexity riterion

Introdu tion

Interiority

Separability index

⊲

Convexity

riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 20

x̄ is k-interior if it is inside the onvex hull of any |X| − k + 1

points of X

- (halfspa e, lo ation, Tukey) depth

onvexity riterion

Introdu tion

Interiority

Separability index

⊲

Convexity

riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 20

x̄ is k-interior if it is inside the onvex hull of any |X| − k + 1

points of X

• is 1-interior

- (halfspa e, lo ation, Tukey) depth

onvexity riterion

Introdu tion

Interiority

Separability index

⊲

Convexity

riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 20

x̄ is k-interior if it is inside the onvex hull of any |X| − k + 1

points of X

• is 1-interior, and 2-interior

- (halfspa e, lo ation, Tukey) depth

onvexity riterion

Introdu tion

Interiority

Separability index

⊲

Convexity

riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 20

x̄ is k-interior if it is inside the onvex hull of any |X| − k + 1

points of X

• is 1-interior, and 2-interior

d(•) is the minimum number of points to be removed from

X su h that • is outside onv. hull of the remaining points

- (halfspa e, lo ation, Tukey) depth

onvexity riterion

Introdu tion

Interiority

Separability index

⊲

Convexity

riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 20

x̄ is k-interior if it is inside the onvex hull of any |X| − k + 1

points of X

• is 1-interior, and 2-interior

d(•) is the minimum number of points to be removed from

X su h that • is outside onv. hull of the remaining points

d(•) = 2

- (halfspa e, lo ation, Tukey) depth

onvexity riterion

Introdu tion

Interiority

Separability index

⊲

Convexity

riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 20

x̄ is k-interior if it is inside the onvex hull of any |X| − k + 1

points of X

• is 1-interior, and 2-interior

d(•) is the minimum number of points to be removed from

X su h that • is outside onv. hull of the remaining points

d(•) = 2

d( ) - (halfspa e, lo ation, Tukey) depth

iso-interiority regions

Introdu tion

Interiority

Separability index

⊲

Convexity

riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 21

iso-interiority regions

Introdu tion

Interiority

Separability index

⊲

Convexity

riterion

Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 21

depth →֒ auto-interiority

Introdu tion

Interiority

Separability index

Convexity riterion

⊲ Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 22

for , is the minimum number of points to be removed

from su h that is outside the onv.hull of the remaining

points

i� is a vertex of the onv.hull

depth of

should distinguishes between on�gurations on entrated in the

"interior� of the onv.hull, and those whi h o ur mainly on the

"margins�...

This is relevant as it would allow dete tion the pattern of the

loud of points issue 2

depth →֒ auto-interiority

Introdu tion

Interiority

Separability index

Convexity riterion

⊲ Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 22

for x ∈ X, d(x) is the minimum number of points to be removed

from X su h that x is outside the onv.hull of the remaining

points

i� is a vertex of the onv.hull

depth of

should distinguishes between on�gurations on entrated in the

"interior� of the onv.hull, and those whi h o ur mainly on the

"margins�...

This is relevant as it would allow dete tion the pattern of the

loud of points issue 2

depth →֒ auto-interiority

Introdu tion

Interiority

Separability index

Convexity riterion

⊲ Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 22

for x ∈ X, d(x) is the minimum number of points to be removed

from X su h that x is outside the onv.hull of the remaining

points

d(x) = 1 i� x is a vertex of the onv.hull(X)

depth of

should distinguishes between on�gurations on entrated in the

"interior� of the onv.hull, and those whi h o ur mainly on the

"margins�...

This is relevant as it would allow dete tion the pattern of the

loud of points issue 2

depth →֒ auto-interiority

Introdu tion

Interiority

Separability index

Convexity riterion

⊲ Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 22

for x ∈ X, d(x) is the minimum number of points to be removed

from X su h that x is outside the onv.hull of the remaining

points

d(x) = 1 i� x is a vertex of the onv.hull(X)

DX = {d(x) : x ∈ X} depth of X

should distinguishes between on�gurations on entrated in the

"interior� of the onv.hull, and those whi h o ur mainly on the

"margins�...

This is relevant as it would allow dete tion the pattern of the

loud of points issue 2

depth →֒ auto-interiority

Introdu tion

Interiority

Separability index

Convexity riterion

⊲ Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 22

for x ∈ X, d(x) is the minimum number of points to be removed

from X su h that x is outside the onv.hull of the remaining

points

d(x) = 1 i� x is a vertex of the onv.hull(X)

DX = {d(x) : x ∈ X} depth of X

should distinguishes between on�gurations on entrated in the

"interior� of the onv.hull, and those whi h o ur mainly on the

"margins�...

This is relevant as it would allow dete tion the pattern of the

loud of points issue 2

depth →֒ auto-interiority

Introdu tion

Interiority

Separability index

Convexity riterion

⊲ Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 22

for x ∈ X, d(x) is the minimum number of points to be removed

from X su h that x is outside the onv.hull of the remaining

points

d(x) = 1 i� x is a vertex of the onv.hull(X)

DX = {d(x) : x ∈ X} depth of X

should distinguishes between on�gurations on entrated in the

"interior� of the onv.hull, and those whi h o ur mainly on the

"margins�...

This is relevant as it would allow dete tion the pattern of the

loud of points

issue 2

depth →֒ auto-interiority

Introdu tion

Interiority

Separability index

Convexity riterion

⊲ Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 22

for x ∈ X, d(x) is the minimum number of points to be removed

from X su h that x is outside the onv.hull of the remaining

points

d(x) = 1 i� x is a vertex of the onv.hull(X)

DX = {d(x) : x ∈ X} depth of X

should distinguishes between on�gurations on entrated in the

"interior� of the onv.hull, and those whi h o ur mainly on the

"margins�...

This is relevant as it would allow dete tion the pattern of the

loud of points →֒ issue 2

if X is uniformly distributed on an n-ball

Introdu tion

Interiority

Separability index

Convexity riterion

⊲ Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 23

Result holds if is uniformly distributed on the losed region of

delimited by an ellipsoid

if X is uniformly distributed on an n-ball

Introdu tion

Interiority

Separability index

Convexity riterion

⊲ Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 23

d(x)

|X|

Result holds if is uniformly distributed on the losed region of

delimited by an ellipsoid

if X is uniformly distributed on an n-ball

Introdu tion

Interiority

Separability index

Convexity riterion

⊲ Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 23

d(x)

|X|

vol( )

vol(Bn)= Vr(x)

Result holds if is uniformly distributed on the losed region of

delimited by an ellipsoid

if X is uniformly distributed on an n-ball

Introdu tion

Interiority

Separability index

Convexity riterion

⊲ Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 23

d(x)

|X|

vol( )

vol(Bn)= Vr(x)

E[Vr(x)] =

∫

Bn

Vr(x)

vol(Bn)dx

Result holds if is uniformly distributed on the losed region of

delimited by an ellipsoid

if X is uniformly distributed on an n-ball

Introdu tion

Interiority

Separability index

Convexity riterion

⊲ Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 23

d(x)

|X|

vol( )

vol(Bn)= Vr(x)

E[Vr(x)] =

∫

Bn

Vr(x)

vol(Bn)dx =

1

2n+1

Result holds if is uniformly distributed on the losed region of

delimited by an ellipsoid

if X is uniformly distributed on an n-ball

Introdu tion

Interiority

Separability index

Convexity riterion

⊲ Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 23

d(x)

|X|

vol( )

vol(Bn)= Vr(x)

E[Vr(x)] =

∫

Bn

Vr(x)

vol(Bn)dx =

1

2n+1

Result holds if X is uniformly distributed on the losed region of

Rn

delimited by an ellipsoid

if X is uniformly distributed on an n-ball

Introdu tion

Interiority

Separability index

Convexity riterion

⊲ Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 24

E[Vr(x)] =1

2n+1

if X is uniformly distributed on an n-ball

Introdu tion

Interiority

Separability index

Convexity riterion

⊲ Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 24

E[Vr(x)] =1

2n+1=⇒

1

|X|

∑x

Vr(x) ≈1

2n+1

if X is uniformly distributed on an n-ball

Introdu tion

Interiority

Separability index

Convexity riterion

⊲ Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 25

(a) (b) (c)

normal uniform ldi

0 1874 0 1247 0 0582

even for small , , for uniformly

distributed

if X is uniformly distributed on an n-ball

Introdu tion

Interiority

Separability index

Convexity riterion

⊲ Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 25

(a) (b) (c)

normal uniform ldi

1

|X|

∑x Vr(x) 0·1874 0·1247 0·0582

even for small , , for uniformly

distributed

if X is uniformly distributed on an n-ball

Introdu tion

Interiority

Separability index

Convexity riterion

⊲ Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 25

(a) (b) (c)

normal uniform ldi

1

|X|

∑x Vr(x) 0·1874 0·1247 0·0582

E[Vr(x)] =1

2n+1=

1

23= 0.125

even for small , , for uniformly

distributed

if X is uniformly distributed on an n-ball

Introdu tion

Interiority

Separability index

Convexity riterion

⊲ Auto-interiority

A general notion of interiority 6th Winter S hool on Te hnology Assessment � 25

(a) (b) (c)

normal uniform ldi

1

|X|

∑x Vr(x) 0·1874 0·1247 0·0582

E[Vr(x)] =1

2n+1=

1

23= 0.125

even for small |X|, 1

|X|

∑x Vr(x) ≈ E[Vr(x)] =

1

2n+1 , for uniformly

distributed X