Transcript 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 # S # S # S # S # S # S S ## S # S #S S ## #S# ##S S # SS # S # S # S # S # S # S # S # S • 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 gG 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) 6000 Z 4000 2000 0 • The core of idea of maxent is: • Find the probability distribution that: 8000 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 i1 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