Maximum Entropyspatial modeling with imperfect

data

Spatial modeling with imperfect data

• Example: species distribution models• What’s wrong with available data and why we

want to use it anyway• Comparison of different approaches• Maximum Entropy approach: Maxent• Model evaluation: AUC

Building models with imperfect data

• Example: constructing (or reconstructing) species distributions

• Paleoecology, conservation, speciation, invasion• Often data is presence-only (Elith et al. 2006):

– Museum records– Herbaria– Fossil locals– Reported sightings

• Sparse data• Spatial bias• Temporal bias• Uncertainty in absence records

Building models with imperfect data

Elith et al. 2006:• Evaluated methods for modeling species

distributions using presence-only data• Compared 16 methods for 226 species in 6

geographic regions• Models were built using presence-only data

and climate and environmental layers• Evaluated against independent presence-

absence datasets

Building models with imperfect data

Elith et al. 2006:• Models that use only occurrences:

– Envelope• BIOCLIM

– Distance based• DOMAIN, LIVES

• Models that characterize background (psuedo-absence):– GLM, GAM, MARS, GARP, MAXENT, BRT, GDM, MARS-

COMM

• Some models implemented in several ways

Building models with imperfect dataElith et al. 2006:

Building models with imperfect dataElith et al. 2006:

Building models with imperfect dataElith et al. 2006:

Maxent: A maximum entropy approach

• Occurrence is a Lat-Long pair denoting location of observation\collection

• Layers that inform the model are from same geographic area in raster format

• Model represents approximation of the realized niche for species

• Assumed that the realized niche and the fundamental niche for the species coincide

• Increasing sampling in larger geographic area (and thus including more variation in environmental conditions encountered by the species) may increase the fraction of fundamental niche represented by occurrences

Fundamental Niche

Fundamental Niche

Realized Niche

Fundamental Niche

Maxent methods• The approximation of an unknown probability distribution should satisfy

any known constraints, and subject to those constraints should have maximum entropy (Jaynes, 1957)

• Maximum Entropy is an epistemic approach to Bayes’ rule• The Monkey Example:

– A team of monkeys are employed to create images by throwing balls at a grid of bins

– Every so often the grid is removed and replaced by a new one– Eventually the monkeys will create multiple copies of each possible

arrangement

Maxent methods• The Monkey Example (cont.):

– Given some evidence about true grid some of the monkey’s grids can be ruled out

– Those left constitute the feasible set, and that which appears most often is a reasonable choice

– Assuming the monkeys are not biased, this choice is consistent with the data but noncommittal about information we do not have

Maxent methods• π is the unknown probability distribution over a

finite set X (the set of pixels or points in the study area)

• The distribution defines a non-negative probability π(x) to each point x

• These probabilities sum to 1• Best approximation of π is the probability

distribution π(hat)• The entropy of π(hat) is:

Maxent methods• Constraints on π(hat) for layers informing model:

– Linear features- continuous variables should be close to their observed values (their mean at occurrence localities)

– Quadratic features- variance of continuous variables should be close to observed values

– Product features- covariance of two continuous variables should be close to observed values

– Threshold feature- proportion of model that has values above a threshold for a continuous variable should be close to observed proportion

– Binary feature- the proportion of each category in a categorical feature should be close to the observed proportions

Maxent methods• Regularization parameter Bj governs how close the

constraints need to match the observed value (without regularization they must be equal)

• Program allows a user-specified proportion of occurrence locals to be reserved from model training for model testing

• absences can be randomly selected (pseudo-absences for presence only) or specified by user (if P-A data available)

• Model will run for either a set number of iterations or until the gain from each iteration falls below a set threshold

Maxent example:brown-throated three-toed sloth, Bradypus variegatus

Maxent example:brown-throated three-toed sloth, Bradypus variegatus

Log contribution of each variable to the raw prediction value

Maxent example:brown-throated three-toed sloth, Bradypus variegatus

Other Maxent Applications

(Siva 1990)

Model EvaluationArea under ROC curve (AUC)

• Receiver Operating Characteristic • Contingency Table:

Actual Value (Data)

Predicted Outcome(Model)

Presence(pos)

Absence(neg)

Presence (pos)

True Positive (TP)

False Positive(FP)

Absence (neg)

False Negative(FN)

True Negative(TN)

Model EvaluationArea under ROC curve (AUC)

• Sensitivity- True Positive Rate (TPR)

Actual Value (Data)

Predicted Outcome(Model)

Presence(pos)

Absence(neg)

Presence (pos)

True Positive (TP)

False Positive(FP)

Absence (neg)

False Negative(FN)

True Negative(TN)

Model EvaluationArea under ROC curve (AUC)

• Specificity- True Negative Rate (TNR)

Actual Value (Data)

Predicted Outcome(Model)

Presence(pos)

Absence(neg)

Presence (pos)

True Positive (TP)

False Positive(FP)

Absence (neg)

False Negative(FN)

True Negative(TN)

Model EvaluationArea under ROC curve (AUC)

• Specificity- True Negative Rate (TNR)• ROC is Sensitivity by (1- Specificity)=(FPR)

Actual Value (Data)

Predicted Outcome(Model)

Presence(pos)

Absence(neg)

Presence (pos)

True Positive (TP)

False Positive(FP)

Absence (neg)

False Negative(FN)

True Negative(TN)

Model EvaluationArea under ROC curve (AUC)

• An example:

• TPR = 63/100 = .63• FPR = 28/100 = .28

Actual Value (Data)

Predicted Outcome(Model)

P=100 N=100

P=91 TP=63 FP=28

N=109 FN=37 TN=72

Image from wikipedia

AP=100 N=100

P=91 TP=63 FP=28

N=109 FN=37 TN=72

BP=100 N=100

P=154 TP=77 FP=77

N=46 FN=23 TN=23

CP=100 N=100

P=112 TP=24 FP=88

N=88 FN=76 TN=12

Image from wikipedia

AP=100 N=100

P=91 TP=63 FP=28

N=109 FN=37 TN=72

BP=100 N=100

P=154 TP=77 FP=77

N=46 FN=23 TN=23

CP=100 N=100

P=112 TP=24 FP=88

N=88 FN=76 TN=12

C’P=100 N=100

P=112 TP=88 FP=24

N=88 FN=12 TN=76

TPTNFP

FN

TPTNFP

FN

TPTN

FN

TPTNFP

FN

TPTN FP

AUC > 0.5 Higher Predictive PowerAUC = 0.5 Random ChanceAUC < 0.5 Worse than Random

• False Positive Rate• True Positive Rate• 1• 1 False Positive Rate• True Positive Rate• 1• 1• 0• 0

• 0• 0

False Positive Rate

Tru

e P

ositi

ve R

ate

1

1

0

0

Model EvaluationArea under ROC curve (AUC)

References:Elith, J., Graham, C. H., Anderson, R. P., Dudı´k, M., Ferrier, S., Guisan, A., Hijmans, R. J.,Huettmann, F.,

Leathwick, J. R., Lehmann, A., Li, J., Lohmann, L. G., Loiselle, B. A., Manion, G.,Moritz, C., Nakamura, M., Nakazawa, Y., Overton, J. McC., Peterson, A. T., Phillips, S. J.,Richardson, K. S., Scachetti-Pereira, R., Schapire, R. E., Sobero´n, J., Williams, S., Wisz, M. S. and Zimmermann, N. E. 2006. Novel methods improve prediction of species’ distributions from occurrence data. Ecography 29: 129 -151

Jaynes, E.T. , 1957. Information theory and statistical mechanics. Phys. Rev. 106, 620 - 630

Lobo, J. M., Jiménez-Valverde, A. and Real R. 2007. AUC a misleading measure of the performance of predictive distribution models. Global Ecology and Biogeography

Phillips, S. J., Dudik, M. and Schapire, R. E. 2004. A maximum entropy approach to species distribution modeling. Proceedings of the 21st International Conference on Machine Learning, Banff, Canada

Phillips, S. J., Anderson, R. P. and Schapire, R. E. 2006. Maximum entropy modeling of species geographic distributions. Ecological Modeling 190: 231-259

Siva, D. S., 1990. Bayesian Inductive Inference Maximum Entropy & Neutron Scattering. Los Alamos Science, Summer: 180 – 206

Maxent program website (its free): http://www.cs.princeton.edu/~schapire/maxent/