2.1. Simulation study

Using simulated data, we compared how alternative random forest approaches ranked and were able to identify true covariates despite data imbalance. First, we generated 1,080 simulated data sets varying in size, imbalance level, number of true variables, their effect sizes, and the strength of multicollinearity between true and noise variables (Figure 1a,b). Based on 121 real-world environmental covariates, we generated simulated binary data. We named the minority class presences and the majority class absences, thus simulating data frequently used in the field of species distribution modeling (Franklin, Reference Franklin2009). From the 121 covariates, we chose covariates with either low or high multicollinearity to other covariates as true variables. To estimate multicollinearity, we calculated variance inflation factors (VIFs, Zuur et al., Reference Zuur, Ieno, Walker, Saveliev and Smith2009). The simulated occurrences were generated by drawing from the Bernoulli distribution with probability p, the probability of the simulated occurrence being a presence. p was calculated as the inverse logit of the linear predictor α + β ₁ × x ₁ + … β_n × x_n, where x _1-n were either one, two, or four true variables with low VIFs or one, two, or four true variables with high VIFs. For each of the six combinations of true variables, we chose regression coefficients β _1-n , that represented weak (β = 0.4), medium (β = 0.9), or strong (β = 1.4) effects, generating 18 simulated responses in total (Figure 1a). The true variables were standardized to a mean of 0 and a standard deviation of 1. Next, we randomly sampled to create 60 data sets from each of the simulated responses (Figure 1b). The data sets varied in sample size (300, 200, 100 absences) and the level of imbalance (20, 15, 10, 5% presences). For each of the 12 combinations of sample size with imbalance level, we randomly sampled five times, generating 1,080 simulated data sets (Figure 1b: 18 simulated responses × 3 sample sizes × 4 imbalance levels × 5 random samples = 1,080).

Figure 1.

Conceptual diagram of the simulation study: (a) Out of a set of 121 environmental covariates, 14 variables were selected as true variables (true var). Seven of the 14 variables had low variance inflation factors (VIFs) indicating low multicollinearity with other covariates and the other seven variables had high variance inflation factors. True variables were standardized to a mean of 0 and a standard deviation of 1. Eighteen binary, simulated response variables were created based on weak (ß = 0.4), medium (ß = 0.9), and strong (ß = 1.4) effect sizes for true variables and either one, two, or four true variables with either low or high VIFs. (b) From each simulated response variable consisting of presences and absences, 60 datasets were created differing in the sample size (300, 200, or 100 absences) and the imbalance level (20, 15, 10, or 5% presences). Presences and absences were randomly drawn five times for each of the 12 combinations of sample size with imbalance level. In total, 1,080 data sets were created (18 simulated responses × 3 sample sizes × 4 imbalance levels × 5 random samples). (c) For each of the 1,080 datasets consisting of simulated presences and absences and the corresponding 121 environmental covariates, variable importances were calculated for each of the 121 covariates using eight alternative random forest versions: four random forest versions were implemented with R package randomForest (RF, RF-Down, RF-Down-Acc, RF-Priors), two versions were implemented with R package ranger (Rang-Hellinger, Rang-Weight) and two versions with R package party (CFor, CFor-AUC): (1) default RF (RF), (2) the majority class downsampled to the size of the minority class (RF-Down), (3) the majority class downsampled to 64% of the size of the minority class (RF-Down-Acc), (4) weights as class priors (RF-Priors), (5) Hellinger distance as splitting criterion (Rang-Hellinger), (6) weights applied to the splitting rule and majority vote (Rang-Weight), (7) random forest based on conditional inference trees (CFor), (8) random forest based on conditional inference trees with AUC permutation importance (CFor-AUC). After the calculation of variable importances, the covariates were ranked from the covariate with the highest importance to the covariate with the lowest importance and the ranks for true variables were extracted. (d) For each of the 1,080 datasets we carried out variable selection using three alternative versions: (1) Variable ranking based on permutation importances from the default RF, followed by a forward selection with the Out-of-Bag error as selection criterion (RF/OOB), (2) Variable ranking based on AUC permutation importance and forests with conditional inference trees, followed by a forward selection with the Out-of-Bag error as selection criterion (CFor-AUC/OOB), and (3) Variable ranking based on AUC permutation importance and forests with conditional inference trees, followed by a forward selection with the AUC as selection criterion (CFor-AUC/AUC).

2.2. Variable ranking

For each of the 1,080 data sets, we evaluated how highly the true predictors were ranked in comparison to the other covariates by eight alternative algorithm versions (Figure 1c). Four random forest versions were implemented with the R package “randomForest” (Liaw and Wiener, Reference Liaw and Wiener2002). This function implements the original programme by Breiman and Cutler (Liaw and Wiener, Reference Liaw and Wiener2002) and forests are based on CART trees (Breiman, Reference Breiman2001; Strobl et al., Reference Strobl, Malley and Tutz2009). CART trees use the Gini split criterion, which measures the decrease in node impurity, to select the predictor for a node split. Subset sampling for individual trees is with replacement by default (Liaw and Wiener, Reference Liaw and Wiener2002). Two random forest versions were implemented with the R package “ranger,” a fast implementation of random forests (Wright and Ziegler, Reference Wright and Ziegler2017). Two versions were implemented with function cforest from R package “party” (Strobl et al., Reference Strobl, Boulesteix, Zeileis and Hothorn2007; Strobl et al. Reference Strobl, Boulesteix, Kneib, Augustin and Zeileis2008). Random forest with cforest are based on conditional inference trees with predictors for node split selected based on a conditional inference independence test (Strobl et al., Reference Strobl, Boulesteix, Zeileis and Hothorn2007). Variable importances based on cforest optionally allow for the calculation of a conditional version of variable importances (Strobl et al., Reference Strobl, Boulesteix, Kneib, Augustin and Zeileis2008). Although this may be suitable for our data, which contained correlated predictor variables, we did not use it to avoid confounding our comparisons with the previous six algorithm versions. The following eight versions were evaluated:

1. 1. RF: Random forest with no adjustment for unbalanced data: function randomForest in package randomForest.

2. 2. RF-Down: For each tree in the ensemble, we randomly deselected data from the majority class to match the size of the minority class using option sampsize in function randomForest.

3. 3. RF-Down-Acc: Downsampling the majority class to the size of the minority class does not necessarily result in the best performance (Ruiz-Gazen and Villa, Reference Ruiz-Gazen and Villa2007). We downsampled the majority class to 64% of the size of the minority class, which had produced good error rates for the minority class on preliminary simulated data.

4. 4. RF-Priors: We provided priors for the classes using the argument classwt of function randomForest (package randomForest). We used the proportion of the absences in the whole data set as the prior for the absence class, and the proportion of presences as the prior for the presence class. To explore the variability in output due to variation in priors, we additionally reversed the order and supplied the proportion of presences as the prior for the absence class and the proportion of absences as the prior for the presence class.

5. 5. Rang-Hellinger: We used the Hellinger distance as the splitting rule in function ranger of package ranger.

6. 6. Rang-Weight: We used the argument class.weights of function ranger to apply weights for the classes in the splitting rule and in the majority voting. We used the proportion of the absences in the whole data set as the weight for the presence class, and the proportion of presences as the weight for the absence class. To explore the variability in output due to variation in weights, we additionally reversed the order and supplied the proportion of presences as the weight for the presence class and the proportion of absences as the weight for the absence class.

7. 7. CFor: We applied random forest based on conditional inference trees with function cforest in package party.

8. 8. CFor-AUC: We calculated AUC permutation importance using function varImpAUC in package “varImp” (Probst, Reference Probst2020) in conjunction with function cforest from package party. Function varImpAUC uses AUC instead of OOB in the calculation of permutation importance (Janitza et al., Reference Janitza, Strobl and Boulesteix2013). The permutation importance is calculated as the difference in AUC before and after permutation of each covariate in turn. Permutation importances are therefore expected to be more insensitive to the imbalance level.

For the tuning parameters of random forest, ntree (the number of trees in a forest) and mtry (the number of variables tried at each node split), we used ntree = 2,000 and mtry = p/2 following Genuer et al. (Reference Genuer, Poggi and Tuleau-Malot2010), where p is the number of predictors. Higher ntree settings result in less variability and more stable variable importances and OOB error (Liaw and Wiener, Reference Liaw and Wiener2002; Díaz-Uriarte and de Andrés, Reference Díaz-Uriarte and de Andrés2006; Genuer et al., Reference Genuer, Poggi and Tuleau-Malot2010). The default value in the R package random forest is ntree = 500 (Liaw and Wiener, Reference Liaw and Wiener2002). With higher mtry values, the permutation importance becomes more conditional; with lower mtry values the permutation importance becomes more marginal (Strobl et al., Reference Strobl, Boulesteix, Kneib, Augustin and Zeileis2008; Grömping, Reference Grömping2009). The default value for classification in the R package random forest is mtry = $ \surd p $ (Liaw and Wiener, Reference Liaw and Wiener2002). For each of the 1,080 test sets and each of the eight algorithms, we repeated the calculation of variable importance measures 10 times. We ranked all 121 covariates based on the average variable importances from the 10 repetitions.

2.3. Variable selection

We evaluated how well variable selection identified true predictors out of all 121 covariates for three algorithm versions. We implemented the variable selection approach of Genuer et al. (Reference Genuer, Poggi and Tuleau-Malot2010), which consists of a forward selection of variables ranked by their variable importances resulting in a subset of important variables inclusive of some redundant variables. Optionally, the redundant variables can be reduced in an additional step, which we did not implement. The variable selection strategy has been described as a prediction-oriented strategy accepting a higher risk of false negatives, or important covariates that are not in the final selection set (Genuer et al., Reference Genuer, Poggi and Tuleau-Malot2015). The variable selection consisted of the following steps for the default version without adjustments to account for unbalanced data: First, all covariates were ranked by their permutation importance, averaged over 50 repetitions. Second, we calculated OOB error for the model with the highest ranked covariate, averaged over 25 repetitions for each set. We added the second highest ranked covariate, recalculated OOB error and repeated this procedure until all covariates were included. Third, we identified the model with the lowest mean OOB error and added its standard deviation. Then we selected the smallest (fewest covariates) model with an OOB error less than this value. We used the tuning parameters suggested by Genuer et al. (Reference Genuer, Poggi and Tuleau-Malot2010): for the calculation of variable importances, we used ntree = 2,000 and mtry = p/2, where p equals the number of covariates and for the calculation of OOB error we used the default ntree and mtry values of function randomForest. To increase computational efficiency we also followed their recommendation for initial removal of the lowest ranked covariates, based on high standard deviations of permutation importances relative to small mean permutation importance per covariate. For full details of the variable selection strategy, see Genuer et al. (Reference Genuer, Poggi and Tuleau-Malot2010).

We evaluated the following three approaches (Figure 1d):

1. 1. RF/OOB: Permutation importances were calculated with the default RF.

2. 2. CFor-AUC/OOB: AUC permutation importances were calculated based on the cforest implementation.

3. 3. CFor-AUC/AUC: AUC permutation importances were calculated based on the cforest implementation of random forest. Instead of using OOB as the selection criterium in the forward selection, AUC was used. In other words, instead of calculating OOB for each nested model, we calculated AUC. Then, analogue to the selection method based on OOB, we identified the model with the highest mean AUC, subtracted its standard deviation from the mean AUC value, then selected the smallest model with a mean AUC larger than this value as the final model (Figure 2).

Figure 2.

Example of model selection using AUC. First, AUC is calculated for all nested models using several repetitions for each nested model. The black curve shows the mean AUC over all repetitions with the gray polygon showing the mean ± standard deviation. Then, the model (vertical black line) with the highest mean AUC (dashed horizontal black line) is identified. The AUC value corresponding to mean–standard deviation for this model (dotted horizontal black line) serves as a threshold accounting for the variation between repetitions. The selected model is the smallest (lowest number of covariates) model with a mean AUC at or above the threshold value (blue lines).

For each selected model, we assessed how well true predictors were identified out of all 121 covariates using sensitivity and specificity. Sensitivity is higher the more true predictors are correctly identified (true positives) and is calculated as:

$$ \frac{\mathrm{true}\ \mathrm{positives}}{\mathrm{true}\ \mathrm{positives}+\mathrm{false}\ \mathrm{negatives}}, $$

where false negatives are true predictors that were not selected. Specificity is higher the more covariates which are not the true predictors are identified as such (true negatives) and was calculated as:

$$ \frac{\mathrm{true}\ \mathrm{negatives}}{\mathrm{false}\ \mathrm{positives}+\mathrm{true}\ \mathrm{negatives}}, $$

where false positives are covariates falsely identified as true predictors (Lin and Chen, Reference Lin and Chen2012).

2.4. Case studies

We present studies from the field of species distribution modeling as case studies. Species occurrence data from sample sites are modeled as a function of environmental information in species distribution models. Subsequently, maps of the occurrence of the species can be produced based on the species-environment associations established in the models (Franklin, Reference Franklin2009). Species distribution modelers are often encouraged to carefully select suitable covariates based on ecological knowledge of the species to avoid spurious associations. However, the ecology of many species is poorly studied and understood (Dormann et al., Reference Dormann, Schymanski, Cabral, Chuine, Graham, Hartig, Kearney, Morin, Römermann, Schröder and Singer2012, Urban et al., Reference Urban, Bocedi, Hendry, Mihoub, Pe’er, Singer, Bridle, Crozier, De Meester, Godsoe, Gonzalez, Hellmann, Holt, Huth, Johst, Krug, Leadley, Palmer, Pantel, Schmitz, Zollner and Travis2016), which carries the risk that important processes are inadvertently omitted or misspecified in models. A key consideration in ecology is that each ecological process has an appropriate spatial scale and that appropriate spatial scales are dependent on the organism under consideration (Wiens, Reference Wiens1976; Addicott et al., Reference Addicott, Aho, Antolin, Padilla, Richardson and Soluk1987). This scale of effect is the scale at which the association between a species occurrence and an environmental covariate is strongest (Jackson and Fahrig, Reference Jackson and Fahrig2012). Another key consideration is that ecological processes can act at multiple spatial scales and influence species at individual, group, or population level (Johnson, Reference Johnson1980; Wiens, Reference Wiens1989; Levin, Reference Levin1992; Cunningham and Johnson, Reference Cunningham and Johnson2006; McGarigal et al., Reference McGarigal, Wan, Zeller, Timm and Cushman2016). A covariate influencing a species’ distribution at one scale (e.g., surface water availability within 500 m of a species’ nest site) may exert a different or no influence at a different scale (e.g., within 10 km; Wiens, Reference Wiens1989, Hamer and Hill, Reference Hamer and Hill2000). Including covariates at appropriate spatial scales in ecological models is therefore important. However, if the spatial process scales are unknown, appropriate spatial scales of covariates have to be selected empirically out of a potentially large pool of covariate × scale combinations. Random forest is a candidate algorithm for such problems due to its ability to assess species-environment associations with all covariate × scale combinations simultaneously in one model (Bradter et al., Reference Bradter, Kunin, Altringham, Thom and Benton2013). Such models may generate new hypotheses about the ecological processes determining the distribution of species.

2.5. Study species and data collection

As case studies, we used the wader species (shorebirds in North America) Northern lapwing (Vannellus vanellus), common snipe (Gallinago gallinago), and common redshank (Tringa totanus). We analyzed and mapped their distribution in a part of the Yorkshire Dales, an upland area of the UK. Habitat selection of the three species is well-studied, particularly at finer spatial scales (e.g., Baines, Reference Baines1990; Green et al., Reference Green, Hirons and Cresswell1990; O’Brien, Reference O’Brien2002; Taylor and Grant, Reference Taylor and Grant2004; Smart et al., Reference Smart, Gill, Sutherland and Watkinson2006; Sharps et al., Reference Sharps, Garbutt, Hiddink, Smart and Skov2016). Therefore, species-environment relationships suggested by our models can be assessed against existing knowledge on the species ecology, while at larger spatial scales, there is potential for the generation of new hypotheses. Specifically, at the territory level, all three species are known to be associated with wet conditions (Green et al., Reference Green, Hirons and Cresswell1990; O’Brien, Reference O’Brien2002; Smart et al., Reference Smart, Gill, Sutherland and Watkinson2006). Northern lapwing is also known to preferred gentle over steep slopes and to avoid improved pastures (Henderson et al., Reference Henderson, Wilson, Steele and Vickery2002; Taylor and Grant, Reference Taylor and Grant2004). Northern lapwing is listed as globally near threatened (IUCN, 2022). In the UK, Northern lapwing is listed as red, and common snipe and common redshank as amber conservation status (Woodward et al., Reference Woodward, Massimino, Hammond, Barber, Barimore, Harris, Leech, Noble, Walker, Baillie and Robinson2020). Managing these and other declining species is a major concern to many land managers.

We surveyed Northern lapwing, common redshank and common snipe in 244 observation units (each ca. 300 × 500 m) in the Yorkshire Dales, UK (Supplementary Figure S1). With observation units of this size we aimed to approximate the reported sizes of core areas used during the breeding season (home ranges henceforth): common redshank chicks were found up to 180 ± 68 m from their nests in salt marshes in Germany (Thyen et al., Reference Thyen, Exo, Cervencl, Esser and Oberdiek2008); Northern lapwing chicks up to 202 m (61–386 m) from nests in Swedish farmland (Johansson and Blomqvist, Reference Johansson and Blomqvist1996) and incubating female common snipe foraged 17–390 m from their nests in lowland sites of the UK (Green et al., Reference Green, Hirons and Cresswell1990). We surveyed the observation units from 61 transects (each 2 km long) to minimize traveling time between observation units, where each transect dissected four contiguous observation units. Surveys targeted elevations below 500 m and were carried out during the 2008 breeding season. To reduce the risk of recording false absences each transect was surveyed three times between April 2 and July 1. This covered much of the incubation and chick-rearing periods of the species (Robinson, Reference Robinson2017). Imperfect detection of species can for example be accounted for in occupancy models (MacKenzie et al., Reference MacKenzie, Nichols, Hines, Knutson and Franklin2003). Modeling approaches that retain the flexibility of machine learning approaches while accounting for imperfect detection are an emerging field (see Mohankumar and Hefley, Reference Mohankumar and Hefley2022 for guidance) and we do not apply these methods here. Each observation unit was recorded as having the focal species present when at least one individual (excluding sightings of flocks) was recorded during at least one survey (for Northern lapwing, due to its earlier laying date relative to common snipe and common redshank, exclusive of the last repeat survey). The data were slightly unbalanced for Northern lapwing and highly unbalanced for common snipe and common redshank: with 37% (90) presences in 244 observation units for Northern lapwing, 18% (44) for common snipe and 8% (19) for common redshank. For a full description of the bird surveys, see Supplementary Appendix S1.

2.6. Environmental covariates

We studied environmental variation at the scale of observation units and coarser scales. Bird species distributions are frequently influenced by food availability (Green et al., Reference Green, Hirons and Cresswell1990; Pearce-Higgins and Yalden, Reference Pearce-Higgins and Yalden2004), microclimate (Wiebe and Martin, Reference Wiebe and Martin1998), habitat structure and composition (Pearce-Higgins and Yalden, Reference Pearce-Higgins and Yalden2004; Chalfoun and Martin, Reference Chalfoun and Martin2009), disturbance (Gill et al., Reference Gill, Sutherland and Watkinson1996), or perceived predation risk (Fontaine and Martin, Reference Fontaine and Martin2006; van der Wal and Palmer, Reference van der Wal and Palmer2008). Distribution data of these variables are rarely available for large areas, so we used more widely available Geographical Information System (GIS) data with credible possible relationships to these possible drivers as proxies. For food availability, we used soil characteristics, elevation, slope, aspect, and rainfall; for microclimatic conditions, we used elevation, aspect, and rainfall; for habitat structure and composition we used soil characteristics, livestock numbers, and satellite data; for wet areas, we used satellite data; for disturbance, we used human settlements, roads, paths, and so forth, and the number of walking groups recorded during surveys; for perceived predation risk we used human settlements, field walls, and viewshed. For a description of the covariates and the possible links between our proxies and food availability, microclimate, habitat structure and composition, disturbance and perceived predation risk, see Supplementary Appendix S1.

Several variables were categorical but we did not always have good prior knowledge of the appropriate grouping for categories. For example, we expected that west-facing areas exposed to the stronger winds (Met Office, 2015) would have a more unfavorable microclimate than south-facing areas receiving more solar irradiation, but did not know if south-west-facing areas should be grouped with the former or the latter. Therefore, we initially created a fine division of categories and checked for possible grouping of neighboring categories after variable selection with random forest (see analysis below).

Boyce et al. (Reference Boyce, Mallory, Morehouse, Prokopenko, Scrafford and Warbington2017) argued that covariates should be measured in a way that is relevant to how the study species perceives the environment. We calculated covariates as the amount of a certain habitat (area or length) within circular buffers of varying size. These values correspond, for example, to the area of wet soil within 0.5 km or the length of paths and roads within 1 km. Our rationale was that birds are highly mobile and therefore the area of, for example, wet soil within a certain distance will be more relevant than the number of patches of wet soil or their size or shape. We chose circular buffers with radii of 0.25, 1, 2.5, 5, 7, and 10 km. Some covariates were used at fewer scales or at a single scale only, such as covariates extracted from satellite imagery. The values of each covariate at each spatial scale were extracted for the centroid of the part of each observation unit visible from the transect. The total number of covariates was 288: 216 multi-scale and 72 single-scale.

2.7. Analysis

For each species, we simultaneously evaluated all single and multi-scale covariates with random forest and the cFor-AUC/AUC variable selection approach outlined above. In other words, we used AUC both in the calculation of permutation importances and as the threshold criterion in variable selection. We used the conditional version of permutation importance where permutations of a covariate are carried out such as to preserve the correlation structure with other covariates, which allows a better distinction between true covariates and covariates correlated to true variables (Strobl et al., Reference Strobl, Boulesteix, Kneib, Augustin and Zeileis2008). Additionally, for the selected covariates, we identified clusters of covariates with Spearman rho > |0.7|, a commonly used threshold (Dormann et al., Reference Dormann, Elith, Bacher, Buchmann, Carl, Carré, García Marquéz, Gruber, Lafourcade, Leitão, Münkemüller, McClean, Osborne, Reineking, Schröder, Skidmore, Zurell and Lautenbach2013), and reduced the covariates to the highest ranked of the cluster. We compared the OOB error with this reduced set to the OOB error before omission of the correlated covariates. As the OOB error remained similar, or even improved slightly (common snipe) we simplified the models by accepting the reduced sets to aid model interpretation.

For the covariates thus selected, we reviewed category groupings. We inspected partial dependence plots of retained covariates and additively grouped neighboring categories at the same spatial scale if the shape of the plots suggested a similar relationship with presence–absence of the species. For Northern lapwing, we additively grouped north-east, east, south-east, and south-facing aspects at the 10 km scale. Partial dependence plots were also used to describe the direction of species–habitat relationships in the final models. To generate predicted maps of the species distribution based on the covariates thus selected, we downsampled the majority class to the number of samples in the minority class to better balance prediction error between the classes.

2.8. Software used

The analyses were carried out in R 4.0.3 and R 4.1.2 (R Core Team, 2020). AUC was calculated using package ROCR (Sing et al., Reference Sing, Sander, Beerenwinkel and Lengauer2005). Processing of GIS data was carried out in ArcGIS 9.2 and 10.3 (ESRI, 2010).