Equivalence of MAXENT and Poisson Point Process Models for Species Distribution Modeling in Ecology

doi:10.1111/J.1541-0420.2012.01824.X

Home
/
Papers
/
Equivalence of MAXENT and Poisson Point Process Models for Species Distribution Modeling in Ecology

Journal Article•DOI•

Equivalence of MAXENT and Poisson Point Process Models for Species Distribution Modeling in Ecology

Ian Renner¹, David I. Warton¹•Institutions (1)

University of New South Wales¹

01 Mar 2013-Biometrics (John Wiley & Sons, Ltd)-Vol. 69, Iss: 1, pp 274-281

TL;DR: It is shown that MAXENT is equivalent to a Poisson regression model and hence is related to aPoisson point process model, differing only in the intercept term, which is scale-dependent in MAXENT.

read less

Abstract: Summary Modeling the spatial distribution of a species is a fundamental problem in ecology. A number of modeling methods have been developed, an extremely popular one being MAXENT, a maximum entropy modeling approach. In this article, we show that MAXENT is equivalent to a Poisson regression model and hence is related to a Poisson point process model, differing only in the intercept term, which is scale-dependent in MAXENT. We illustrate a number of improvements to MAXENT that follow from these relations. In particular, a point process model approach facilitates methods for choosing the appropriate spatial resolution, assessing model adequacy, and choosing the LASSO penalty parameter, all currently unavailable to MAXENT. The equivalence result represents a significant step in the unification of the species distribution modeling literature.

...read moreread less

Citations

PDF

Open Access

More filters

Journal Article•DOI•

Statistics for Spatial Data.

[...]

Andrew B. Lawson¹, Noel A Cressie•Institutions (1)

University of Dundee¹

01 Mar 1993-The Statistician

6,278 citations

Journal Article•DOI•

A practical guide to MaxEnt for modeling species' distributions: what it does, and why inputs and settings matter

[...]

Cory Merow¹, Cory Merow², Matthew J. Smith², John A. Silander¹•Institutions (2)

University of Connecticut¹, Microsoft²

01 Oct 2013-Ecography

TL;DR: A detailed explanation of how MaxEnt works and a prospectus on modeling options are provided to enable users to make informed decisions when preparing data, choosing settings and interpreting output to highlight the need for making biologically motivated modeling decisions.

...read moreread less

Abstract: The MaxEnt software package is one of the most popular tools for species distribution and environmental niche modeling, with over 1000 published applications since 2006. Its popularity is likely for two reasons: 1) MaxEnt typically outperforms other methods based on predictive accuracy and 2) the software is particularly easy to use. MaxEnt users must make a number of decisions about how they should select their input data and choose from a wide variety of settings in the software package to build models from these data. The underlying basis for making these decisions is unclear in many studies, and default settings are apparently chosen, even though alternative settings are often more appropriate. In this paper, we provide a detailed explanation of how MaxEnt works and a prospectus on modeling options to enable users to make informed decisions when preparing data, choosing settings and interpreting output. We explain how the choice of background samples reflects prior assumptions, how nonlinear functions of environmental variables (features) are created and selected, how to account for environmentally biased sampling, the interpretation of the various types of model output and the challenges for model evaluation. We demonstrate MaxEnt’s calculations using both simplified simulated data and occurrence data from South Africa on species of the flowering plant family Proteaceae. Throughout, we show how MaxEnt’s outputs vary in response to different settings to highlight the need for making biologically motivated modeling decisions.

...read moreread less

2,370 citations

Cites background or methods from "Equivalence of MAXENT and Poisson P..."

...1) are a general class models that predict count data (i.e. the number of individuals; Warton and Shepherd 2010, Aarts et al. 2012, Fithian and Hastie 2012, Renner and Warton 2012, Royle et al. 2012)....
[...]
...These two sampling assumptions (individuals vs grid cells) can lead to similar results when presences are spatially sparse but are incompatible when multiple counts are likely to occur per grid cell (Renner and Warton 2012)....
[...]
...This approach corresponds to treating MaxEnt as a traditional statistical model (cf. Renner and Warton 2012)....
[...]
...Furthermore, Renner and Warton (2012) note that the assumption of spatial independence of PO samples may be frequently be violated, but that inhomogeneous Poisson point process models can remedy this to some extent....
[...]
...(2) is a discretized approximation to more general class of inhomogeneous Poisson point process models that treat the landscape as continuous (Warton and Shepherd 2010, Chakraborty et al. 2011, Fithian and Hastie 2012, Renner and Warton 2012)....
[...]

Journal Article•DOI•

Opening the black box: an open-source release of Maxent

[...]

Steven J. Phillips¹, Robert P. Anderson¹, Robert P. Anderson², Miroslav Dudík³, Robert E. Schapire³, Mary E. Blair¹ - Show less +2 more•Institutions (3)

American Museum of Natural History¹, City University of New York², Microsoft³

01 Jul 2017-Ecography

TL;DR: A new open-source release of the Maxent software for modeling species distributions from occurrence records and environmental data is announced, and a new R package for fitting Maxent models using the glmnet package for regularized generalized linear models is described.

...read moreread less

Abstract: This software note announces a new open-source release of the Maxent software for modeling species distributions from occurrence records and environmental data, and describes a new R package for fitting such models. The new release (ver. 3.4.0) will be hosted online by the American Museum of Natural History, along with future versions. It contains small functional changes, most notably use of a complementary log-log (cloglog) transform to produce an estimate of occurrence probability. The cloglog transform derives from the recently-published interpretation of Maxent as an inhomogeneous Poisson process (IPP), giving it a stronger theoretical justification than the logistic transform which it replaces by default. In addition, the new R package, maxnet, fits Maxent models using the glmnet package for regularized generalized linear models. We discuss the implications of the IPP formulation in terms of model inputs and outputs, treating occurrence records as points rather than grid cells and interpreting the exponential Maxent model (raw output) as as an estimate of relative abundance. With these two open-source developments, we invite others to freely use and contribute to the software.

...read moreread less

1,343 citations

Cites background or methods from "Equivalence of MAXENT and Poisson P..."

...Because Maxent is an IPP, standard generalized linear modeling software can be used to fit Maxent models via Poisson regression (Renner and Warton 2013), or even more conveniently, using standard logistic regression (Fithian and Hastie 2013)....
[...]
...…logistic regression (IWLR): the maxnet package Because Maxent is an IPP, standard generalized linear modeling software can be used to fit Maxent models via Poisson regression (Renner and Warton 2013), or even more conveniently, using standard logistic regression (Fithian and Hastie 2013)....
[...]
...More recently, it was noted that the exact same maximum likelihood exponential model can be obtained from an inhomogeneous Poisson process (IPP) (Aarts et al. 2012, Fithian and Hastie 2013, Renner and Warton 2013)....
[...]

Journal Article•DOI•

ENMeval: An R package for conducting spatially independent evaluations and estimating optimal model complexity for Maxent ecological niche models

[...]

Robert Muscarella¹, Peter J. Galante², Mariano Soley-Guardia², Robert A. Boria², Jamie M. Kass², María Uriarte¹, Robert P. Anderson², Robert P. Anderson³ - Show less +4 more•Institutions (3)

Columbia University¹, City University of New York², American Museum of Natural History³

01 Nov 2014-Methods in Ecology and Evolution

TL;DR: ENMeval as mentioned in this paper is an R package that creates data sets for k-fold cross-validation using one of several methods for partitioning occurrence data (including options for spatially independent partitions), builds a series of candidate models using Maxent with a variety of user-defined settings and provides multiple evaluation metrics to aid in selecting optimal model settings.

...read moreread less

Abstract: Summary Recent studies have demonstrated a need for increased rigour in building and evaluating ecological niche models (ENMs) based on presence-only occurrence data. Two major goals are to balance goodness-of-fit with model complexity (e.g. by ‘tuning’ model settings) and to evaluate models with spatially independent data. These issues are especially critical for data sets suffering from sampling bias, and for studies that require transferring models across space or time (e.g. responses to climate change or spread of invasive species). Efficient implementation of procedures to accomplish these goals, however, requires automation. We developed ENMeval, an R package that: (i) creates data sets for k-fold cross-validation using one of several methods for partitioning occurrence data (including options for spatially independent partitions), (ii) builds a series of candidate models using Maxent with a variety of user-defined settings and (iii) provides multiple evaluation metrics to aid in selecting optimal model settings. The six methods for partitioning data are n−1 jackknife, random k-folds ( = bins), user-specified folds and three methods of masked geographically structured folds. ENMeval quantifies six evaluation metrics: the area under the curve of the receiver-operating characteristic plot for test localities (AUCTEST), the difference between training and testing AUC (AUCDIFF), two different threshold-based omission rates for test localities and the Akaike information criterion corrected for small sample sizes (AICc). We demonstrate ENMeval by tuning model settings for eight tree species of the genus Coccoloba in Puerto Rico based on AICc. Evaluation metrics varied substantially across model settings, and models selected with AICc differed from default ones. In summary, ENMeval facilitates the production of better ENMs and should promote future methodological research on many outstanding issues.

...read moreread less

1,210 citations

Cites methods from "Equivalence of MAXENT and Poisson P..."

...Recent work has demonstrated equivalency between the MAXENT algorithm and loglinear generalized linear models (Renner & Warton 2013), as well as close links to inhomoge- neous Poisson process (IPP) models (Fithian & Hastie 2013)....
[...]

Journal Article•DOI•

Making better Maxent models of species distributions: complexity, overfitting and evaluation

[...]

Aleksandar Radosavljevic¹, Robert P. Anderson¹, Robert P. Anderson²•Institutions (2)

City University of New York¹, American Museum of Natural History²

01 Apr 2014-Journal of Biogeography

TL;DR: In this paper, the authors integrate solutions to these issues for Maxent models, using the Caribbean spiny pocket mouse, Heteromys anomalus, as an example, by selecting appropriate evaluation data, detecting overfitting and tuning program settings to approximate optimal model complexity.

...read moreread less

Abstract: Aim Models of species niches and distributions have become invaluable to biogeographers over the past decade, yet several outstanding methodological issues remain. Here we address three critical ones: selecting appropriate evaluation data, detecting overfitting, and tuning program settings to approximate optimal model complexity. We integrate solutions to these issues for Maxent models, using the Caribbean spiny pocket mouse, Heteromys anomalus, as an example. Location North-western South America. Methods We partitioned data into calibration and evaluation datasets via three variations of k-fold cross-validation: randomly partitioned, geographically structured and masked geographically structured (which restricts background data to regions corresponding to calibration localities). Then, we carried out tuning experiments by varying the level of regularization, which controls model complexity. Finally, we gauged performance by quantifying discriminatory ability and overfitting, as well as via visual inspections of maps of the predictions in geography. Results Performance varied among data-partitioning approaches and among regularization multipliers. The randomly partitioned approach inflated estimates of model performance and the geographically structured approach showed high overfitting. In contrast, the masked geographically structured approach allowed selection of high-performing models based on all criteria. Discriminatory ability showed a slight peak in performance around the default regularization multiplier. However, regularization levels two to four times higher than the default yielded substantially lower overfitting. Visual inspection of maps of model predictions coincided with the quantitative evaluations. Main conclusions Species-specific tuning of model parameters can improve the performance of Maxent models. Further, accurate estimates of model performance and overfitting depend on using independent evaluation data. These strategies for model evaluation may be useful for other modelling methods as well.

...read moreread less

1,051 citations

Cites methods from "Equivalence of MAXENT and Poisson P..."

...More generally, results based on the tuning approach should be compared with model selection based on information criteria (e.g. the AIC corrected for small sample size, AICc; Warren & Seifert, 2011) and generalized cross validation (GCV), which is similar in intent (Renner & Warton, 2013)....
[...]
...…as the Akaike information criterion (AIC) or nonlinear generalized cross validation (GCV) – which each penalize increasingly complex models – but recently proposed use of such approaches for ecological niche models requires further empirical testing (Warren & Seifert, 2011; Renner & Warton, 2013)....
[...]

1
2
3
4
…
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78

Collapse

References

PDF

Open Access

More filters

Journal Article•DOI•

Regression Shrinkage and Selection via the Lasso

[...]

Robert Tibshirani

01 Jan 1996-Journal of the royal statistical society series b-methodological

TL;DR: A new method for estimation in linear models called the lasso, which minimizes the residual sum of squares subject to the sum of the absolute value of the coefficients being less than a constant, is proposed.

...read moreread less

Abstract: SUMMARY We propose a new method for estimation in linear models. The 'lasso' minimizes the residual sum of squares subject to the sum of the absolute value of the coefficients being less than a constant. Because of the nature of this constraint it tends to produce some coefficients that are exactly 0 and hence gives interpretable models. Our simulation studies suggest that the lasso enjoys some of the favourable properties of both subset selection and ridge regression. It produces interpretable models like subset selection and exhibits the stability of ridge regression. There is also an interesting relationship with recent work in adaptive function estimation by Donoho and Johnstone. The lasso idea is quite general and can be applied in a variety of statistical models: extensions to generalized regression models and tree-based models are briefly described.

...read moreread less

40,785 citations

"Equivalence of MAXENT and Poisson P..." refers background or methods or result in this paper

...Our analysis will consist of four steps: (1) determine the appropriate spatial resolution for analysis; (2) assess whether a Poisson point process model is appropriate; (3) estimate the LASSO parameter (Tibshirani, 1996) for regularization; and (4) compare results with a MAXENT model....
[...]
...Alternatively, some datadriven criterion could be used to try to choose a λ which optimizes predictive performance (Tibshirani, 1996; Fu, 2005; Zou, Hastie, and Tibshirani, 2007)....
[...]
...Prior to applying the LASSO to point process models, variables were standardized to have mean 0 and variance 1 as in Tibshirani (1996), such that the LASSO penalty was applied to standardized coefficients....
[...]

Book•

Generalized Linear Models

[...]

Peter McCullagh¹, John A. Nelder•Institutions (1)

Imperial College London¹

01 Jan 1983

TL;DR: In this paper, a generalization of the analysis of variance is given for these models using log- likelihoods, illustrated by examples relating to four distributions; the Normal, Binomial (probit analysis, etc.), Poisson (contingency tables), and gamma (variance components).

...read moreread less

Abstract: The technique of iterative weighted linear regression can be used to obtain maximum likelihood estimates of the parameters with observations distributed according to some exponential family and systematic effects that can be made linear by a suitable transformation. A generalization of the analysis of variance is given for these models using log- likelihoods. These generalized linear models are illustrated by examples relating to four distributions; the Normal, Binomial (probit analysis, etc.), Poisson (contingency tables) and gamma (variance components).

...read moreread less

23,215 citations

Journal Article•DOI•

Maximum entropy modeling of species geographic distributions

[...]

Steven J. Phillips¹, Robert P. Anderson², Robert P. Anderson³, Robert E. Schapire⁴•Institutions (4)

AT&T Labs¹, City University of New York², American Museum of Natural History³, Princeton University⁴

25 Jan 2006-Ecological Modelling

TL;DR: In this paper, the use of the maximum entropy method (Maxent) for modeling species geographic distributions with presence-only data was introduced, which is a general-purpose machine learning method with a simple and precise mathematical formulation.

...read moreread less

13,120 citations

"Equivalence of MAXENT and Poisson P..." refers background in this paper

...MAXENT (Phillips, Anderson, and Schapire, 2006), based on a maximum entropy approach, is particularly common in SDM, having been cited 378 times in 2011 according to Google Scholar....
[...]

Journal Article•DOI•

Information Theory and Statistical Mechanics. II

[...]

E. T. Jaynes¹•Institutions (1)

Stanford University¹

15 Oct 1957-Physical Review

TL;DR: In this article, the authors consider statistical mechanics as a form of statistical inference rather than as a physical theory, and show that the usual computational rules, starting with the determination of the partition function, are an immediate consequence of the maximum-entropy principle.

...read moreread less

Abstract: Information theory provides a constructive criterion for setting up probability distributions on the basis of partial knowledge, and leads to a type of statistical inference which is called the maximum-entropy estimate. It is the least biased estimate possible on the given information; i.e., it is maximally noncommittal with regard to missing information. If one considers statistical mechanics as a form of statistical inference rather than as a physical theory, it is found that the usual computational rules, starting with the determination of the partition function, are an immediate consequence of the maximum-entropy principle. In the resulting "subjective statistical mechanics," the usual rules are thus justified independently of any physical argument, and in particular independently of experimental verification; whether or not the results agree with experiment, they still represent the best estimates that could have been made on the basis of the information available.It is concluded that statistical mechanics need not be regarded as a physical theory dependent for its validity on the truth of additional assumptions not contained in the laws of mechanics (such as ergodicity, metric transitivity, equal a priori probabilities, etc.). Furthermore, it is possible to maintain a sharp distinction between its physical and statistical aspects. The former consists only of the correct enumeration of the states of a system and their properties; the latter is a straightforward example of statistical inference.

...read moreread less

12,099 citations

"Equivalence of MAXENT and Poisson P..." refers background in this paper

...Its rise in popularity has been meteoric, having only been introduced to ecology 6 years ago, although the concept of maximum entropy modeling has been around for a long time (Jaynes, 1957)....
[...]

Book•

Generalized Linear Models

[...]

John H. Schuenemeyer, Peter McCullagh, John A. Nelder

25 Aug 2003

10,034 citations