the maximum diversity of a compact metric space
TRANSCRIPT
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The Maximum Diversity of a Compact Metric Space
Emily RoffThe University of Edinburgh
Magnitude WorkshopICMS, Edinburgh
4th July 2019
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Objectives
Part I: Measuring diversity
• Introduce a family of distance-sensitive entropies (Dq)q∈[0,∞].
Part II: Maximising diversity on compact spaces
• State a maximum entropy theorem for compact spaces.
Part II: Maximum diversity and magnitude
• Define a real-valued invariant, closely related to magnitude.
Part III: Uniform measures• Propose a uniform measure for a wide class of metric spaces.
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Objectives
Part I: Measuring diversity
• Introduce a family of distance-sensitive entropies (Dq)q∈[0,∞].
Part II: Maximising diversity on compact spaces
• State a maximum entropy theorem for compact spaces.
Part II: Maximum diversity and magnitude
• Define a real-valued invariant, closely related to magnitude.
Part III: Uniform measures• Propose a uniform measure for a wide class of metric spaces.
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Objectives
Part I: Measuring diversity
• Introduce a family of distance-sensitive entropies (Dq)q∈[0,∞].
Part II: Maximising diversity on compact spaces
• State a maximum entropy theorem for compact spaces.
Part II: Maximum diversity and magnitude
• Define a real-valued invariant, closely related to magnitude.
Part III: Uniform measures• Propose a uniform measure for a wide class of metric spaces.
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Objectives
Part I: Measuring diversity
• Introduce a family of distance-sensitive entropies (Dq)q∈[0,∞].
Part II: Maximising diversity on compact spaces
• State a maximum entropy theorem for compact spaces.
Part II: Maximum diversity and magnitude
• Define a real-valued invariant, closely related to magnitude.
Part III: Uniform measures• Propose a uniform measure for a wide class of metric spaces.
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Provenance
• Leinster and Cobbold, Measuring diversity: the importance ofspecies similarity. Ecology 93 (2012).
• Leinster and Meckes, Maximizing diversity in biology and beyond.Entropy 18(3) (2016).
• Meckes, Magnitude, diversity, capacities, and dimensions of metricspaces. Potential Analysis 42(2) (2015).
• Leinster and Roff, The maximum diversity of a compact space.(To appear.)
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Part I
Measuring Diversity
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Modelling a community
Conventionally, an ecological community is modelled by a set of species and aprobability distribution recording the relative abundance of each species.
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Quantifying diversity
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A single family of entropies
DefinitionLet X = {1, . . . , n}, let µ = (µ1, . . . , µn) be a probability distributionon X , and let q ∈ [0,∞]. The Renyi entropy of order q of µ is
Hq(µ) =1
1− qlog
∑i∈supp(µ)
µqi
for q 6= 1,∞, and
Hq(µ) =
{−∑
i∈supp(µ) µi logµi when q = 1,
− log maxi∈supp(µ) µi when q =∞.
In particular, H1(µ) is Shannon entropy.
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Incorporating similarities between species
Now, let X = {1, . . . n} be a set of species, with pairwise similaritiesrecorded in an n× n matrix K . Let µ be a probability distribution on X .
DefinitionFor q ∈ [0,∞) not equal to 1, the order-q diversity of µ is
DKq (µ) =
(∑i
(Kµ)iq−1µi
) 11−q
.
When K is the identity matrix, this reduces to Renyi entropy:
D Iq(µ) = exp(Hq(µ)).
Moral:log(diversity) = generalised entropy.
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Incorporating similarities between species
Now, let X = {1, . . . n} be a set of species, with pairwise similaritiesrecorded in an n× n matrix K . Let µ be a probability distribution on X .
DefinitionFor q ∈ [0,∞) not equal to 1, the order-q diversity of µ is
DKq (µ) =
(∑i
(Kµ)iq−1µi
) 11−q
.
When K is the identity matrix, this reduces to Renyi entropy:
D Iq(µ) = exp(Hq(µ)).
Moral:log(diversity) = generalised entropy.
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Interpreting diversity
For q ∈ [0,∞) not equal to 1, the order-q diversity of µ is
DKq (µ) =
(∑i
(Kµ)iq−1µi
) 11−q
.
The number (Kµ)i =∑
j Kijµj is the expected similarity between anindividual of species i and an individual chosen at random—it is thetypicality of species i .
Then (Kµ)i−1 is the atypicality of species i .
So DKq (µ) is the average atypicality of species in X with respect to µ,
where ‘average’ refers to the power mean of order 1− q.
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Interpreting diversity
For q ∈ [0,∞) not equal to 1, the order-q diversity of µ is
DKq (µ) =
(∑i
(Kµ)iq−1µi
) 11−q
.
The number (Kµ)i =∑
j Kijµj is the expected similarity between anindividual of species i and an individual chosen at random—it is thetypicality of species i .
Then (Kµ)i−1 is the atypicality of species i .
So DKq (µ) is the average atypicality of species in X with respect to µ,
where ‘average’ refers to the power mean of order 1− q.
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Interpreting diversity
For q ∈ [0,∞) not equal to 1, the order-q diversity of µ is
DKq (µ) =
(∑i
(Kµ)iq−1µi
) 11−q
.
The number (Kµ)i =∑
j Kijµj is the expected similarity between anindividual of species i and an individual chosen at random—it is thetypicality of species i .
Then (Kµ)i−1 is the atypicality of species i .
So DKq (µ) is the average atypicality of species in X with respect to µ,
where ‘average’ refers to the power mean of order 1− q.
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Interpreting diversity
For q ∈ [0,∞) not equal to 1, the order-q diversity of µ is
DKq (µ) =
(∑i
(Kµ)iq−1µi
) 11−q
.
The number (Kµ)i =∑
j Kijµj is the expected similarity between anindividual of species i and an individual chosen at random—it is thetypicality of species i .
Then (Kµ)i−1 is the atypicality of species i .
So DKq (µ) is the average atypicality of species in X with respect to µ,
where ‘average’ refers to the power mean of order 1− q.
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Part II
Maximising Diversity on Compact Spaces
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Compact spaces with similarities
DefinitionLet X be a compact Hausdorff topological space.
A similarity kernel on X is a continuous function K : X × X → [0,∞)satisfying K (x , x) > 0 for all x ∈ X .
The pair (X ,K ) is a space with similarities.
We say (X ,K ) is symmetric if K (x , y) = K (y , x) for all x , y ∈ X .
Examples
• A set of species with a similarity matrix.
• A compact metric space X with kernel K (x , y) = e−d(x ,y).
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Compact spaces with similarities
DefinitionLet X be a compact Hausdorff topological space.
A similarity kernel on X is a continuous function K : X × X → [0,∞)satisfying K (x , x) > 0 for all x ∈ X .
The pair (X ,K ) is a space with similarities.
We say (X ,K ) is symmetric if K (x , y) = K (y , x) for all x , y ∈ X .
Examples
• A set of species with a similarity matrix.
• A compact metric space X with kernel K (x , y) = e−d(x ,y).
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Typicality for compact spaces
DefinitionLet (X ,K ) be a space with similarities, and let P(X ) be the space ofRadon probability measures on X , with the weak* topology.
For each µ ∈ P(X ) and x ∈ X , define
(Kµ)(x) =
∫K (x ,−) dµ ∈ [0,∞).
The function Kµ : X → [0,∞) is the typicality function of (X ,K , µ).
The atypicality function of (X ,K , µ) is 1/Kµ.
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Diversity for compact spaces
DefinitionLet (X ,K ) be a space with similarities, µ ∈ P(X ), and q ∈ [0,∞) notequal to 1. The diversity of order q of µ is
DKq (µ) =
(∫(Kµ)q−1 dµ
)1/(1−q).
The diversity function of order q of (X ,K ) is DKq : P(X )→ (0,∞).
Example
Let X be a compact metric space, and µ ∈ P(X ). Then
Dq(µ) =
(∫ (∫e−d(x ,y) dµ(x)
)q−1dµ(y)
)1/(1−q)
.
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Diversity for compact spaces
DefinitionLet (X ,K ) be a space with similarities, µ ∈ P(X ), and q ∈ [0,∞) notequal to 1. The diversity of order q of µ is
DKq (µ) =
(∫(Kµ)q−1 dµ
)1/(1−q).
The diversity function of order q of (X ,K ) is DKq : P(X )→ (0,∞).
Example
Let X be a compact metric space, and µ ∈ P(X ). Then
Dq(µ) =
(∫ (∫e−d(x ,y) dµ(x)
)q−1dµ(y)
)1/(1−q)
.
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The viewpoint parameter matters!
Leinster and Cobbold, Measuring Diversity. . . , Ecology 93 (2012)
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A maximisation theorem
Theorem (Leinster and Roff, 2019)
Let (X ,K ) be a nonempty symmetric space with similarities.
1. There exists a probability measure µ on X that maximises DKq (µ)
for all q ∈ [0,∞] at once.
2. The maximum diversity supµDKq (µ) is independent of q ∈ [0,∞].
DefinitionThe maximum diversity of (X ,K ) is
Dmax(X ) = supµ
DKq (µ)
for any q ∈ [0,∞].
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A maximisation theorem
Theorem (Leinster and Roff, 2019)
Let (X ,K ) be a nonempty symmetric space with similarities.
1. There exists a probability measure µ on X that maximises DKq (µ)
for all q ∈ [0,∞] at once.
2. The maximum diversity supµDKq (µ) is independent of q ∈ [0,∞].
DefinitionThe maximum diversity of (X ,K ) is
Dmax(X ) = supµ
DKq (µ)
for any q ∈ [0,∞].
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A maximisation theorem
Theorem (Leinster and Roff, 2019)
Let (X ,K ) be a nonempty symmetric space with similarities.
1. There exists a probability measure µ on X that maximises DKq (µ)
for all q ∈ [0,∞] at once.
2. The maximum diversity supµDKq (µ) is independent of q ∈ [0,∞].
DefinitionThe maximum diversity of (X ,K ) is
Dmax(X ) = supµ
DKq (µ)
for any q ∈ [0,∞].
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Maximum diversity of metric spaces
Example
The maximum diversity of a compact metric space X is
Dmax(X ) = supµ∈P(X )
D2(µ) = supµ∈P(X )
1∫ ∫e−d(x ,y) dµ(x) dµ(y)
.
This quantity is investigated in Meckes (2015).
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Part III
Maximum Diversity and Magnitude
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Relating diversity to magnitude
Let (X ,K ) be a space with similarities.
DefinitionA measure µ ∈ P(X ) is balanced if Kµ is constant on supp µ.
Proposition
Any diversity-maximising measure is balanced.
DefinitionA weight measure on (X ,K ) is a signed measure µ such that Kµ ≡ 1.
For example, a weight measure on a compact metric space satisfies∫e−d(x ,y) dµ(x) = 1 for all y ∈ X .
Corollary
If µ is a maximising measure, its restriction to supp µ is a scalarmultiple of a weight measure.
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Relating diversity to magnitude
Let (X ,K ) be a space with similarities.
DefinitionA measure µ ∈ P(X ) is balanced if Kµ is constant on supp µ.
Proposition
Any diversity-maximising measure is balanced.
DefinitionA weight measure on (X ,K ) is a signed measure µ such that Kµ ≡ 1.
For example, a weight measure on a compact metric space satisfies∫e−d(x ,y) dµ(x) = 1 for all y ∈ X .
Corollary
If µ is a maximising measure, its restriction to supp µ is a scalarmultiple of a weight measure.
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Relating diversity to magnitude
Let (X ,K ) be a space with similarities.
DefinitionA measure µ ∈ P(X ) is balanced if Kµ is constant on supp µ.
Proposition
Any diversity-maximising measure is balanced.
DefinitionA weight measure on (X ,K ) is a signed measure µ such that Kµ ≡ 1.
For example, a weight measure on a compact metric space satisfies∫e−d(x ,y) dµ(x) = 1 for all y ∈ X .
Corollary
If µ is a maximising measure, its restriction to supp µ is a scalarmultiple of a weight measure.
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Relating diversity to magnitude
Let (X ,K ) be a space with similarities.
DefinitionA measure µ ∈ P(X ) is balanced if Kµ is constant on supp µ.
Proposition
Any diversity-maximising measure is balanced.
DefinitionA weight measure on (X ,K ) is a signed measure µ such that Kµ ≡ 1.
For example, a weight measure on a compact metric space satisfies∫e−d(x ,y) dµ(x) = 1 for all y ∈ X .
Corollary
If µ is a maximising measure, its restriction to supp µ is a scalarmultiple of a weight measure.
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Relating diversity to magnitude
DefinitionLet (X ,K ) be a symmetric space with similarities. Suppose there existsa weight measure on (X ,K ). The magnitude of (X ,K ) is
|X | = µ(X )
for any weight measure µ on (X ,K ).
LemmaLet µ be a maximising measure on (X ,K ). Then
Dmax(X ) = |supp µ|.
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Relating diversity to magnitude
DefinitionLet (X ,K ) be a symmetric space with similarities. Suppose there existsa weight measure on (X ,K ). The magnitude of (X ,K ) is
|X | = µ(X )
for any weight measure µ on (X ,K ).
LemmaLet µ be a maximising measure on (X ,K ). Then
Dmax(X ) = |supp µ|.
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Positive definite spaces
Let X be a compact, positive definite metric space. Its magnitude is
|X | = sup{|Y | : finite Y ⊆ X}.
Proposition (Meckes, 2011)
If X admits a weight measure µ, this is equivalent to |X | = µ(X ).
Consequences
• For all Y ⊆ X , we have |Y | ≤ |X |. So when X 6= ∅,
Dmax(X ) ≤ |X |.
• If X admits a nonnegative weight measure µ, its normalisationµ ∈ P(X ) is the unique maximising measure on X , and
Dmax(X ) = |X |.
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Positive definite spaces
Let X be a compact, positive definite metric space. Its magnitude is
|X | = sup{|Y | : finite Y ⊆ X}.
Proposition (Meckes, 2011)
If X admits a weight measure µ, this is equivalent to |X | = µ(X ).
Consequences
• For all Y ⊆ X , we have |Y | ≤ |X |. So when X 6= ∅,
Dmax(X ) ≤ |X |.
• If X admits a nonnegative weight measure µ, its normalisationµ ∈ P(X ) is the unique maximising measure on X , and
Dmax(X ) = |X |.
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Example
A line segment [0, r ] ⊂ R has weight measure 12(δ0 + λ[0,r ] + δr ).
Hence,
Dmax([0, r ]) = |[0, r ]| = 1 +r
2
and the unique maximising measure on [0, r ] is
δ0 + λ[0,r ] + δr
2 + r.
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Example
Suppose X is homogeneous. Then the Haar probability measure µ is theunique maximising measure on X and, for any y ∈ X ,
Dmax(X ) = |X | =1∫
e−d(x ,y) dµ(x).
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Part IV
Uniform Measures
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Defining a uniform measure
Let X = (X , d) be a metric space. Write tX for the space (X , td).
DefinitionLet X be a compact metric space. Suppose that tX has a uniquemaximising measure µt for all t � 0, and that limt→∞ µt exists inP(X ). Then the uniform measure on X is
µX = limt→∞
µt .
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Defining a uniform measure
Let X = (X , d) be a metric space. Write tX for the space (X , td).
DefinitionLet X be a compact metric space. Suppose that tX has a uniquemaximising measure µt for all t � 0, and that limt→∞ µt exists inP(X ). Then the uniform measure on X is
µX = limt→∞
µt .
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Example
Proposition
On a finite metric space, the uniform measure is the uniform measure.
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Example
Proposition
On a homogeneous space, the uniform measure is the Haar measure.
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Example
Proposition
On a compact subinterval of R, the uniform measure is Lebesguemeasure restricted and normalised.
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Example
Proposition
Let X ⊂ Rn be a compact subset with nonzero volume. Let λX beLebesgue measure restricted to X and normalised. For all q ∈ [0,∞],
De−td
q (λX )
Dmax(tX )→ 1 as t →∞.
Moral: λX is approximately maximising at large scales.
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Objectives
Part I: Measuring diversity
• Introduce a family of distance-sensitive entropies (Dq)q∈[−∞,∞].
Part II: Maximising diversity on compact spaces
• State a maximum entropy theorem for compact spaces.
Part II: Maximum diversity and magnitude
• Define a real-valued invariant, closely related to magnitude.
Part III: Uniform measures• Propose a uniform measure for a wide class of metric spaces.