1. On Maxent
Jorge Soberon
University of Kansas

2. 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

3. 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

4. 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…

5. And the data points…
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

6. The guts of Maxent I. The core of idea of maxent is:
Find the probability distribution that:
1) Have the same means of features as the observed means
2) It is as flat as possible (maximizes entropy)

7. 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

8. 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

9. GARP
E-SPACE
MAXENT
BIOCLIM