Download On Maxent

On Maxent
Jorge Soberon
University of Kansas
The idea
• There is an unknown
probability
distribution, denoted
by p.
• The probabilities are
defined over the grid of
cells G
• Probability of what?
Probability of pixel g
being suitable for the
species
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• The values of p(g) are
probabilities, therefore,
they add to 1, and since,
in general, |G| is normally
large, for example, 105 to
107, then the probabilities
tend to be small.
• We wish to estimate p.
Our estimate is called
pˆ
p (g)  0
 p (g)  1
gG
The “features”
• Maxent assumes that for
each cell g in G, there
are “features” that give
a continuous value per
cell:
• f1(g),f2(g),… fn(g)
• Features are average
temperature, minimum
temperature, total
precipitation, elevation,
and so on…
6000
Z
4000
2000
0
• We also have a number
of data points, meaning
the observations.
• Those datapoints define
the mean value of the
features
• That is, we take the mean
value of each feature,
taken over the values in
the cells where the
species was observed
8000
And the data points…
2
1
y
0
Y
p [ y]
4
-1
-2
2
X
0
p [ x]
x
– 1) Have the same means
of features as the
observed means
– 2) It is as flat as possible
(maximizes entropy)
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Z
4000
2000
0
• The core of idea of
maxent is:
• Find the probability
distribution that:
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The guts of Maxent I.
2
1
y
0
Y
p [ y]
4
-1
-2
2
X
0
p [ x]
x
The guts of Maxent II.
• Mathematical arguments
shows that the
Maximum Entropy
distribution will be a
Gibbs distribution
• To prevent
“overfitting”, there are a
regularization factors
• Minimize
• Subject to
pˆ 
e
λ f ( x)
Z
 f ( x)
ei1 i i
v

Z
pˆ[ fi ]  p [ fi ]   j

 v
1 m
ln  Z    λf ( x j )    i i
m j 1

 i 1
q [ f1 ]  p [ f1 ]
.
.
q [ f m ]  p [ f m ]
The output of Maxent
• The estimated probabilities
of suitability for every patch
• An accumulated value
which increases the
numbers.
• Means that if we randomly
sample pixels, t% of them
will have
• A(x)  t
pˆ ( x )
A( x)  100
p
pˆ ( x )  ˆ ( x )
pˆ ( x)
300
300
4000
6000
8000
-100
GARP
10000
-200
2000
0
Temperature
0
100
200
200
100
0
Temperature
-100
E-SPACE
Precipitation
0
2000
4000
6000
8000
10000
100
0
Temperature
100
0
-100
MAXENT
-200
-100
BIOCLIM
-200
Temperature
200
200
300
300
Precipitation
0
2000
4000
6000
Precipitation
8000
10000
0
2000
4000
6000
Precipitation
8000
10000