Maximum Entropy applications for modeling spatial data

Transcript Maximum Entropy applications for modeling spatial data

Maximum Entropy spatial 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 data

Elith et al. 2006:

Building models with imperfect data

Elith et al. 2006:

Building models with imperfect data

Elith et al. 2006:

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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

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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

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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 B

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

Area under ROC curve (AUC) • An example:

Predicted Outcome (Model)

P=91 N=109

Actual Value (Data)

P=100 TP=63 FN=37 N=100 FP=28 TN=72 • • TPR = 63/100 = .63

FPR = 28/100 = .28

P=91 N=109 P=100 TP=63 FN=37 N=100 FP=28 TN=72

P=154 N=46 P=100 TP=77 FN=23 N=100 FP=77 TN=23

P=112 N=88 P=100 TP=24 FN=76 N=100 FP=88 TN=12 Image from wikipedia

P=91 N=109 P=100 TP=63 FN=37 N=100 FP=28 TN=72

P=154 N=46 P=100 TP=77 FN=23 N=100 FP=77 TN=23

P=112 N=88 P=100 TP=24 FN=76 N=100 FP=88 TN=12

C’

P=112 N=88 P=100 TP=88 FN=12 N=100 FP=24 TN=76 Image from wikipedia

TN FP TP FN

FN FP FN

FN FP

Model Evaluation

Area under ROC curve (AUC) • • • • • • • • • 1 1 0 0 True Positive Rate 1 1 False Positive Rate True Positive Rate 0 • • 0 0 0 False Positive Rate AUC > 0.5 Higher Predictive Power AUC = 0.5 Random Chance AUC < 0.5 Worse than Random 1

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 21

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/