Calculate AICc from Maxent model prediction

This function calculates AICc for Maxent models based on Warren and Seifert (2011).

Usage

aic.maxent(p.occs, ncoefs, p = NULL)

Arguments

p.occs: data frame: raw (maxent.jar) or exponential (maxnet) predictions for the occurrence localities based on one or more models
ncoefs: numeric: number of non-zero model coefficients
p: RasterStack: raw (maxent.jar) or exponential (maxnet) model predictions; if NULL, AICc will be calculated based on the background points, which already have predictions that sum to 1 and thus need no correction -- this assumes that the background points represent a good sample of the study extent

Value

data frame with three columns: AICc is the Akaike Information Criterion corrected for small sample sizes calculated as: $$ (2 * K - 2 * logLikelihood) + (2 * K) * (K+1) / (n - K - 1)$$

where K is the number of non-zero coefficients in the model and n is the number of occurrence localities. The logLikelihood is calculated as: $$ sum(log(vals))$$

where vals is a vector of Maxent raw/exponential values at occurrence localities and the sum of these values across the study extent is equal to 1. delta.AICc is the difference between the AICc of a given model and the AICc of the model with the lowest AICc. w.AICc is the Akaike weight (calculated as the relative likelihood of a model (exp(-0.5 * delta.AICc)) divided by the sum of the likelihood values of all models included in a run. These can be used for model averaging (Burnham and Anderson 2002).

Details

As motivated by Warren and Seifert (2011) and implemented in ENMTools (Warren et al. 2010), this function calculates the small sample size version of Akaike Information Criterion for ENMs (Akaike 1974). We use AICc (instead of AIC) regardless of sample size based on the recommendation of Burnham and Anderson (1998, 2004). The number of coefficients is determined by counting the number of non-zero coefficients in the maxent lambda file (m@lambdas for maxent.jar and m$betas for maxnet. See Warren et al. (2014) for limitations of this approach, namely that the number of non-zero coefficients is an estimate of the true degrees of freedom. For Maxent ENMs, AICc is calculated by first standardizing the raw predictions such that all cells in the study extent sum to 1, then extracting the occurrence record predictions. The predictions of the study extent may not sum to 1 if the background does not cover every grid cell -- as the background predictions sum to 1 by definition, extra predictions for grid cells not in the training data will add to this sum. When no raster data is provided, the raw predictions of the occurrence records are used to calculate AICc without standardization, with the assumption that the background records have adequately represented the occurrence records. The standardization is not necessary here because the background predictions sum to 1 already, and the occurrence data is a subset of the background. This will not be true if the background does not adequately represent the occurrence records, in which case the occurrences are not a subset of the background and the raster approach should be used instead. The likelihood of the data for a given model is then calculated by taking the product of the raw occurrence predictions (Warren and Seifert 2011), or the sum of their logs, as is implemented here.

Note

Returns all NAs if the number of non-zero coefficients is larger than the number of observations (occurrence localities).

References

Akaike, H. (1974) A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19: 716-723. https://doi.org/10.1109/TAC.1974.1100705

Burnham, K. P. and Anderson, D. R. (1998) Model selection and multimodel inference: a practical information-theoretic approach. Springer, New York.

Burnham, K. P. and Anderson, D. R. (2004) Multimodel inference: understanding AIC and BIC in model selection. Sociological Methods and Research, 33: 261-304. https://doi.org/10.1177/0049124104268644

Warren, D. L., Glor, R. E, and Turelli, M. (2010) ENMTools: a toolbox for comparative studies of environmental niche models. Ecography, 33: 607-611. https://doi.org/10.1111/j.1600-0587.2009.06142.x

Warren, D. L., & Seifert, S. N. (2011). Ecological niche modeling in Maxent: the importance of model complexity and the performance of model selection criteria. Ecological Applications, 21: 335-342. https://doi.org/10.1890/10-1171.1

Warren, D. L., Wright, A. N., Seifert, S. N., and Shaffer, H. B. (2014). Incorporating model complexity and sampling bias into ecological niche models of climate change risks faced by 90 California vertebrate species of concern. Diversity and Distributions, 20: 334-343. https://doi.org/10.1111/ddi.12160