Magnitude and other measures of metric spaces
Simon Willerton
University of Sheffield
Exploratory meeting on
the mathematics of biodiversity
CRM Barcelona
July 2012
Overview
category
theory
1/15
Overview
category
theory
Leinster
sets with
distance
1/15
Overview
category
theory
geometry
Leinster
Leinster
Willerton
Meckes
sets with
distance
1/15
Overview
category
theory
geometry
Leinster
Willerton
Meckes
Leinster
sets with
distance
Leinster
Cobbold
diversity
measures
1/15
Overview
category
theory
geometry
Leinster
Willerton
Meckes
Leinster
sets with
distance
Leinster
Cobbold
diversity
measures
1/15
“set with distances” = “metric space”
We have
I
a set of ‘points’
I
some notion of distance 0 6 dij 6 ∞ between the ith and jth points.
2/15
“set with distances” = “metric space”
We have
I
a set of ‘points’
I
some notion of distance 0 6 dij 6 ∞ between the ith and jth points.
For example:
NP
Q
S
SP
2/15
“set with distances” = “metric space”
We have
I
a set of ‘points’
I
some notion of distance 0 6 dij 6 ∞ between the ith and jth points.
For example:
NP
6
6
12
Q
S
12
6
6
SP
2/15
“set with distances” = “metric space”
We have
I
a set of ‘points’
I
some notion of distance 0 6 dij 6 ∞ between the ith and jth points.
For example:
NP
6
6
12
Q
S
12
6
6
SP
Note: not every metric space can be thought of as points in Euclidean space.
2/15
Magnitude [Leinster]
Metric space X with similarity matrix Zij := e −dij .
3/15
Magnitude [Leinster]
Metric space X with similarity matrix Zij := e −dij .
Define ‘weight’ (if possible) −∞ < wi < ∞ at each point i so that
X
Zij wj = 1
for every j
j
3/15
Magnitude [Leinster]
Metric space X with similarity matrix Zij := e −dij .
Define ‘weight’ (if possible) −∞ < wi < ∞ at each point i so that
X
Zij wj = 1
for every j
j
1
0.001
1
3/15
Magnitude [Leinster]
Metric space X with similarity matrix Zij := e −dij .
Define ‘weight’ (if possible) −∞ < wi < ∞ at each point i so that
X
Zij wj = 1
for every j
j
0.37
0.73
1
0.001
1
0.37
3/15
Magnitude [Leinster]
Metric space X with similarity matrix Zij := e −dij .
Define ‘weight’ (if possible) −∞ < wi < ∞ at each point i so that
X
Zij wj = 1
for every j
j
0.37
0.73
1
0.001
1
0.37
Define the magnitude by
|X | =
X
wi
i
3/15
Magnitude [Leinster]
Metric space X with similarity matrix Zij := e −dij .
Define ‘weight’ (if possible) −∞ < wi < ∞ at each point i so that
X
Zij wj = 1
for every j
j
0.37
0.73
1
0.001
1
0.37
Define the magnitude by
|X | =
X
|X | ∼ 1.47
wi
i
3/15
Magnitude [Leinster]
Metric space X with similarity matrix Zij := e −dij .
Define ‘weight’ (if possible) −∞ < wi < ∞ at each point i so that
X
Zij wj = 1
for every j
j
0.37
0.73
1
0.001
1
0.37
Define the magnitude by
|X | =
X
|X | ∼ 1.47
wi
i
If Zij is invertible then |X | =
P
(Z −1 )ij .
ij
3/15
Example of scaling
1000t
t
1000t
4/15
Example of scaling
1000t
t
1000t
3
|X |
2
1
0
0.0001 0.001 0.01
0.1
t
1
10
100
4/15
Example of scaling
1000t
t
1000t
3
|X |
2
1
0
0.0001 0.001 0.01
0.1
t
1
10
100
4/15
Example of scaling
1000t
t
1000t
3
|X |
2
1
0
0.0001 0.001 0.01
0.1
t
1
10
100
As any space X is scaled bigger and bigger |X | → N.
4/15
Example of bad metric space
t
t
t
t
5/15
Example of bad metric space
t
5
4
t
3
t
|X |
t
2
1
0
t
0.01
0.1
1
10
100
5/15
Example of bad metric space
t
5
4
t
3
t
|X |
t
2
1
0
t
0.01
0.1
1
10
100
Many metric spaces are better behaved than this.
5/15
Example of bad metric space
t
5
4
t
3
t
|X |
t
2
1
0
t
0.01
0.1
1
10
100
Many metric spaces are better behaved than this.
If Z is positive definite then |X | is defined.
For example, if X is a subset of Euclidean space then |X | is defined.
5/15
Diversity measures [Leinster, Cobbold]
Model our community using
I
a metric space X with similarity matrix Zij
I
a probability (or relative abundance) pi at the ith point.
6/15
Diversity measures [Leinster, Cobbold]
Model our community using
I
a metric space X with similarity matrix Zij
I
a probability (or relative abundance) pi at the ith point.
Effective number of species:

1
1−q
X

q−1


pi (Z p)i




i:pi >0



Y
q Z
i
D (p) :=
(Z p)−p
i



i:pi >0






1

 min
i:pi >0 (Z p)i
q 6= 1,
q = 1,
q = ∞.
6/15
Diversity measures [Leinster, Cobbold]
Model our community using
I
a metric space X with similarity matrix Zij
I
a probability (or relative abundance) pi at the ith point.
Effective number of species:
q
D Z (p)
25
20
15
10
5
0
0
1
2
q
6/15
Diversity measures [Leinster, Cobbold]
Model our community using
I
a metric space X with similarity matrix Zij
I
a probability (or relative abundance) pi at the ith point.
Effective number of species:
q
D Z (p)
25
20
15
10
5
0
0
1
2
q
Recover various other measures of diversity using this.
For example, obtain Hill numbers when dij = ∞ (i.e. Zij = 0) for i 6= j.
6/15
Leinster’s maximazing result
Theorem
Let X be a symmetric metric space. So Z is symmetric.
7/15
Leinster’s maximazing result
Theorem
Let X be a symmetric metric space. So Z is symmetric.
I
If Z is positive definite and there is a weighting with non-negative
weights (wi > 0), then
Dmax (Z ) = |X |
i.e., the magnitude is the maximum diversity for all q, and normalizing
the weights gives the maximizing probability distribution
wi
pi := P
wi
7/15
Leinster’s maximazing result
Theorem
Let X be a symmetric metric space. So Z is symmetric.
I
If Z is positive definite and there is a weighting with non-negative
weights (wi > 0), then
Dmax (Z ) = |X |
i.e., the magnitude is the maximum diversity for all q, and normalizing
the weights gives the maximizing probability distribution
wi
pi := P
wi
I
Otherwise
Dmax (Z ) =
max
Y ⊂X &wi >0
|Y |.
7/15
Summary of magnitude |X |
I
Mathematically natural (if mysterious), c.f. category theory.
I
Related to biodiversity.
I
Seemingly related to geometry in Euclidean space.
I
Can behave rather weirdly at times.
8/15
Other size measures of metric spaces
I
I
Get Hill Numbers by giving a probability space a dull metric.
Get numbers for a metric space by giving a dull probability distribution.
9/15
Other size measures of metric spaces
I
I
Get Hill Numbers by giving a probability space a dull metric.
Get numbers for a metric space by giving a dull probability distribution.
q
E (X ) := q D Z
1
1
N,..., N
9/15
Other size measures of metric spaces
I
I
Get Hill Numbers by giving a probability space a dull metric.
Get numbers for a metric space by giving a dull probability distribution.
q
E (X ) := q D Z
1
1
N,..., N
For example, analogue of species richness:
0
E (X ) :=
N X
N
X
i=1
−1
Zij
j=1
9/15
Other size measures of metric spaces
I
I
Get Hill Numbers by giving a probability space a dull metric.
Get numbers for a metric space by giving a dull probability distribution.
q
E (X ) := q D Z
1
1
N,..., N
For example, analogue of species richness:
0
E (X ) :=
N X
N
X
i=1
−1
Zij
j=1
Note: this is not the same as
|X | =
N X
N
X
(Z )−1
ij
i=1 j=1
9/15
Example of scaling II
1000t
t
1000t
10/15
Example of scaling II
1000t
t
1000t
3
2
1
|X |
0
0.0001 0.001 0.01
0.1
t
1
10
100
10/15
Example of scaling II
1000t
t
1000t
3
2
|X |
E0 (X )
1
0
0.0001 0.001 0.01
0.1
t
1
10
100
10/15
Example of bad metric space II
t
t
t
t
11/15
Example of bad metric space II
5
t
4
t
3
t
2
t
1
0
t
0.01
0.1
1
10
100
11/15
Example of bad metric space II
5
t
4
t
3
t
2
t
1
0
t
0.01
0.1
1
10
100
11/15
Example of bad metric space II
5
t
4
t
3
t
2
t
1
0
t
0.01
0.1
1
I
The size 0 E (X ) is defined for all metric spaces.
I
As X is scaled up 0 E (X ) increases from 1 to N.
I
It is much easier to calculate 0 E (X ) than |X |.
10
100
11/15
Zooming in on a space with 6400 points
12/15
Zooming in on a space with 6400 points
12/15
Zooming in on a space with 6400 points
12/15
Zooming in on a space with 6400 points
12/15
Zooming in on a space with 6400 points
12/15
Zooming in on a space with 6400 points
12/15
Zooming in on a space with 6400 points
12/15
Zooming in on a space with 6400 points
12/15
Dimension
In a metric space we can scale all the distances.
What should happen to the size?
13/15
Dimension
In a metric space we can scale all the distances.
What should happen to the size?
For example, double the distances:
13/15
Dimension
In a metric space we can scale all the distances.
What should happen to the size?
For example, double the distances:
13/15
Dimension
In a metric space we can scale all the distances.
What should happen to the size?
For example, double the distances:
13/15
Dimension
In a metric space we can scale all the distances.
What should happen to the size?
For example, double the distances:
13/15
Dimension
In a metric space we can scale all the distances.
What should happen to the size?
For example, double the distances:
2 times as big
13/15
Dimension
In a metric space we can scale all the distances.
What should happen to the size?
For example, double the distances:
2 times as big
13/15
Dimension
In a metric space we can scale all the distances.
What should happen to the size?
For example, double the distances:
2 times as big
13/15
Dimension
In a metric space we can scale all the distances.
What should happen to the size?
For example, double the distances:
2 times as big
13/15
Dimension
In a metric space we can scale all the distances.
What should happen to the size?
For example, double the distances:
2 times as big
4 times as big
13/15
Dimension
In a metric space we can scale all the distances.
What should happen to the size?
For example, double the distances:
21 = 2 times as big
22 = 4 times as big
13/15
Dimension
In a metric space we can scale all the distances.
What should happen to the size?
For example, double the distances:
21 = 2 times as big
22 = 4 times as big
Think of dimension as how the size changes when the distances are changed.
13/15
Dimension
In a metric space we can scale all the distances.
What should happen to the size?
For example, double the distances:
21 = 2 times as big
22 = 4 times as big
Think of dimension as how the size changes when the distances are changed.
Given ‘size’ can see if it gives a good idea of dimension.
13/15
Size of rectangles with 6400 points
6400
0E
1000
100
10
10 × 640
1
0.00001
0.001
0.1
interpoint distance
10
14/15
Size of rectangles with 6400 points
6400
0E
1000
100
10 × 640
80 × 80
1 × 6400
10
1
0.00001
0.001
0.1
interpoint distance
10
14/15
Rectangles with 6400 points and ‘dimension’
2
growth rate of 0 E
10 × 640
1
0
0.00001
0.001
0.1
interpoint distance
10
15/15
Rectangles with 6400 points and ‘dimension’
growth rate of 0 E
2
10 × 640
80 × 80
1 × 6400
1
0
0.00001
0.001
0.1
interpoint distance
10
15/15
Rectangles with 6400 points and ‘dimension’
growth rate of 0 E
2
10 × 640
80 × 80
1 × 6400
1
0
0.00001
0.001
0.1
interpoint distance
10
There is geometric information is 0 E (X ).
15/15