








key: cord-009797-8mdie73v
authors: Valle, Denis; Albuquerque, Pedro; Zhao, Qing; Barberan, Albert; Fletcher, Robert J.
title: Extending the Latent Dirichlet Allocation model to presence/absence data: A case study on North American breeding birds and biogeographical shifts expected from climate change
date: 2018-08-26
journal: Glob Chang Biol
DOI: 10.1111/gcb.14412
sha: 
doc_id: 9797
cord_uid: 8mdie73v

Understanding how species composition varies across space and time is fundamental to ecology. While multiple methods having been created to characterize this variation through the identification of groups of species that tend to co‐occur, most of these methods unfortunately are not able to represent gradual variation in species composition. The Latent Dirichlet Allocation (LDA) model is a mixed‐membership method that can represent gradual changes in community structure by delineating overlapping groups of species, but its use has been limited because it requires abundance data and requires users to a priori set the number of groups. We substantially extend LDA to accommodate widely available presence/absence data and to simultaneously determine the optimal number of groups. Using simulated data, we show that this model is able to accurately determine the true number of groups, estimate the underlying parameters, and fit with the data. We illustrate this method with data from the North American Breeding Bird Survey (BBS). Overall, our model identified 18 main bird groups, revealing striking spatial patterns for each group, many of which were closely associated with temperature and precipitation gradients. Furthermore, by comparing the estimated proportion of each group for two time periods (1997–2002 and 2010–2015), our results indicate that nine (of 18) breeding bird groups exhibited an expansion northward and contraction southward of their ranges, revealing subtle but important community‐level biodiversity changes at a continental scale that are consistent with those expected under climate change. Our proposed method is likely to find multiple uses in ecology, being a valuable addition to the toolkit of ecologists.

occur in space and time. For example, in a spatial context, these approaches have attempted to identify geographic areas with similar taxa, areas that have been variously called "biogeographical regions" (Gonzales-Orozco, Thornhill, Knerr, Laffan, & Miller, 2014) , "bioregions" (Bloomfield et al., 2017) , or "biogeographical modules" (Carstensen et al., 2012) . Such bioregions have been argued to be important for understanding the role of history on community assemblages (Carstensen et al., 2012 (Carstensen et al., , 2013 , interpreting ecological dynamics (Economo et al., 2015) , and developing broad-scale conservation strategies (Vilhena & Antonelli, 2015) .

The Latent Dirichlet Allocation (LDA; not to be confused with linear discriminant analysis) model is a powerful model-based method to decompose species assemblage data into groups of species that tend to co-occur in space and/or time. The benefits of using this model include the ability to adequately represent uncertainty, accommodate missing data, and, perhaps most importantly, to describe sampling units as comprised of multiple groups (i.e., mixedmembership [MM] units) (Valle, Baiser, Woodall, & Chazdon, 2014) .

Conceptually, the ability to describe sampling units as comprised of multiple groups has rarely been considered in previous methods (i.e., prior approaches are typically based on "hard" partitions) but may better honor community dynamics and may better characterize impacts of environmental change. For instance, biome transition zones, ecotones, and habitat edges are locations that are often comprised of a mix of species groups, providing sources for potentially novel species interactions (Gosz, 1993; Ries, Fletcher, Battin, & Sisk, 2004) . Similarly, climate change is predicted to cause geographic shifts in species and communities, leading to the hypothesis of novel assemblages arising across space as climate and habitat changes (Urban et al., 2016; Williams & Jackson, 2007) . In addition, most partitioning methods that delineate biogeographical regions or modules based on hard boundaries can lead to high uncertainty in boundary delineation-an issue that can be rectified by allowing groups to overlap. It is important to note that LDA allows for overlapping groups, but it does not require it to be present (i.e., if data do not support overlap, no overlap is estimated).

It is unfortunate that the LDA model, as currently developed, has been restricted to abundance data, which are often not available because accurate quantification of abundance can be very challenging and costly. In the absence of abundance data, researchers often have to rely on presence/absence data to understand species distributions and biodiversity patterns (Jones, 2011; Joseph, Field, Wilcox, & Possingham, 2006) . Another limitation of the LDA model is that the number of groups has to be prespecified, requiring researchers to run LDA multiple times to then use some criterion (e.g., AIC) to choose the optimal number of groups (e.g., Valle et al., 2014) , an approach that often can be computationally costly.

In this article, we substantially develop the LDA model to be able to fit the much more commonly available presence/absence data and to automatically determine the optimal number of groups. We start by describing our statistical model. Then, using simulated data, we show how our method automatically detects the optimal number of groups in the data, reliably estimates the underlying parameters, and better fits the data, outperforming other approaches. At last, we illustrate the novel insights gained using our method by analyzing a long-term dataset collected on breeding birds in the United States and Canada (Breeding Bird Survey [BBS]; Pardieck, Ziolkowski, Lutmerding, Campbell, & Hudson, 2017) to determine how environmental variables influence bird assemblages across the continent and how these assemblages are changing through time.

The overall goal of our method is to identify the major patterns of species co-occurrence in the data, each of which we define to be a distinct species group. We adopt the term species group (instead of "bioregion" or other related terms) because these major co-occurrence patterns do not have to have a strong spatial pattern (although they often do), these groups can overlap in space, and proportion of groups can change through time. More specifically, our method characterizes each sampling unit l in terms of the proportion of the different groups (parameter vector θ l ) and characterizes each group k in terms of the probability of the different species (parameter vector ϕ k ). For example, θ l ¼ 0:1; 0:8; 0:1; 0 ½ indicates that the second group dominates unit l and that the fourth group is absent. This example also illustrates that a given sampling unit can be comprised of multiple groups, which explains why these types of models are called mixed-membership models. In the same way, ϕ k ¼ 0; 0:8; 0:8 ½ indicates that species 2 and 3 (but not species 1) are important species of group k. Note that a given species can have a high probability in more than one group. A more formal description of the statistical model is given below.

The data consist of a matrix filled with binary variables x isl (i.e., equal to one if species s was present in observation i and unit l and equal to zero otherwise). Notice that we assume that multiple observations might have been made for each species s and unit l, possibly due to temporally repeated measures or because multiple subsamples were measured within each unit l (e.g., a forest plot comprised of four subplots). Each of these binary variables has an associated latent group membership status z isl . This variable indicates to which group species s in sampling unit l during observation i comes from.

We assume that each observation x isl comes from the following distribution, given that species s in unit l during observation i comes

where ϕ sk is the probability of observing species s if this species came from group k. Notice that z isl influences the distribution for x isl by determining the k subscript of the parameter ϕ. Next, we assume that the latent variable z isl comes from a multinomial distribution:

where θ l is a vector of probabilities that sum to one, and each element θ lk consists of the probability of a species in unit l to have come from group k.

In relation to the priors for our parameters, we adopt a conjugate beta prior for ϕ sk :

Throughout this article, we assume vague priors by setting a and b to 1. Building on the work of Dunson (2010) and Valle et al. (2017) , we adopt a truncated stick-breaking prior for θ l . This prior assumes that:

for k = 1,…,K−1 and γ > 0. We set the parameter for the last group to 1 (i.e., V lK ¼ 1). With these parameters, we calculate θ lk using the

Under this prior, θ lk is a priori stochastically exponentially decreasing as long as γ < 1 and smaller γ tend to enforce greater sparseness (i.e., the existence of fewer groups). For most of the examples in this article, γ was set to 0.1, which we have found to work well for multiple datasets. More details regarding this prior can be found in Supporting Information Appendix S1.

The benefit of this prior is that, if the data support fewer groups than specified by the user, it will tend to force these superfluous additional groups to be empty or to have very few latent variables z isl assigned to them, as illustrated in the simulation section below. This prior also helps to avoid label switching, a common problem in mixed-membership and mixture models. Bayesian Markov chain Monte Carlo (MCMC) algorithms applied to these types of models sometimes mix poorly and can lead to nonsensical results if posterior distributions of parameters are summarized by their averages (Stephens, 2000) . The label switching problem refers to the fact that the labels of the different groups can change (e.g., groups 1 and 2 can become groups 2 and 1, respectively) without changing the likelihood (i.e., the group labels are unidentifiable). Our truncated stick-breaking prior helps to avoid the label switching problem by enforcing an ordering of the groups according to their overall proportions.

We fit the LDA using a Gibbs sampler. A more complete description of this model and the derivation of the full conditional distributions used within this Gibbs sampler are provided in Supporting Information Appendix S1. Supporting Information Appendix S2 contains a short tutorial describing how to fit the model using the code that we make publicly available, reproducing some of the simulated data results.

There are three important points regarding LDA that need to be emphasized. First, the proposed model can accommodate negative and positive correlations between species. To illustrate this, assume that there are just two species groups and two species, s and s'.

Negative correlation between these species is captured by our

. These parameter estimates indicate that, whenever a site has a high proportion of group 1, species s will have a high probability of occurring, whereas species s' will tend to be absent. In the same way, whenever a site has a high proportion of group 2, species s' will have a high probability of occurring but species s will tend to be absent, resulting in negative correlation. Positive correlation between these species is captured by our model if, for example,φ s ¼ 0:95 0:1 ! and ϕ s 0 ¼ 0:8 0:05 ! . These parameter estimates imply that, whenever a site has a high proportion of group 1, both species s and s' will have a high probability of occurring. In the same way, whenever a site has a high proportion of group 2, both species s and s' will have a high probability of being absent, inducing positive correlation.

Second, hard clustering methods that group locations with similar species composition (e.g., Kreft & Jetz, 2010 ) correspond in our model to vectors θ l comprised of zeroes except for a single element which is equal to 1. In the same way, hard clustering methods that group species that tend to co-occur (e.g., Azeria et al., 2009) from a species composition perspective. This is due to the fact that the probability of observing species s for two locations p and q is

(see Supporting Information Appendix S1 for details). In this scenario, the algorithm might determine that a single species group dominates all locations instead of distinguishing the different species groups.

We simulate data to evaluate the performance of the proposed model and to compare its results to those from other clustering methods. To avoid the identifiability problems described above, we generate parameters for all simulations such that each group completely dominates at least one location and that each group has at least one species that is never present in the other groups (ensuring distinct species composition of these groups).

We illustrate with simulated data how the truncated stick-breaking prior can identify the optimal number of groups and how our algorithm can retrieve the true parameter values under a wide range of conditions. More specifically, the true number of groups K* was set to 3 and 10; the number of sampling units (i.e., locations) was set to 100 and 1000; the number of species was set to 50 and 200; and the number of observations per location was set to 5. Parameters were drawn randomly (i.e., ϕ sk ∼ Beta 0:5; 0:5 ð Þ and θ l ∼ Dirichlet 0:1 ð Þ), and the identifiability assumptions described above were then imposed. We adopted a Beta 0:5; 0:5 ð Þdistribution for ϕ sk because this distribution is likely to generate species groups that are more dissimilar in terms of species composition given that it is a U-shaped symmetric distribution. We generated 10 datasets for each combination of these settings, totaling 80 datasets.

To fit these data, we assume a maximum of 20 groups (K = 20) and estimate the true number of groups K* by determining the number of groups that are not superfluous. Superfluous groups are defined to be those groups that are very uncommon across the entire region (i.e., θ lk <0:5 for 99% of the locations, where θ lk is the mean of the posterior distribution). At last, we test the sensitivity of the modeling results to the prior by fitting these data with γ set to 0.1 and 1.

We also compare LDA to other methods using simulated data. In these simulations, we assume data are available on 200 species over 1,000 locations, with five repeated observations per location. Furthermore, 3, 5, and 7 groups were used to generate these data.

Because the goal is to compare inference from different methods, we set the parameters θ lk in such a way that it allows for a straightforward visual appraisal of the advantages and limitations of the different methods. On the other hand, the parameters ϕ ks were randomly drawn from Beta 0:5; 0:5 ð Þ ; and subsequently, the assumption regarding groups with distinct species composition was imposed.

When fitting LDA, we set the maximum number of groups to 20 and rely on the truncated stick-breaking prior with γ ¼ 0:1 to uncover the correct number of groups. We compare and contrast inference from our model to that from competing approaches, including traditional hard clustering methods (i.e., hierarchical and kmean clustering) and mixture models (i.e., region of common profile (RCP) model; Foster et al., 2017; Lyons, Foster, & Keith, 2017 To determine whether these breeding bird groups have been shifting their spatial distribution over time, we divided our study period into two 6-year periods: 1997-2002 and 2010-2015. Each route × period combination resulted in a distinct "sampling unit" (i.e., distinct row in our data matrix), and data from individual years within each time period were treated as repeated observations.

To relate the spatial distribution of the identified groups to potential environmental drivers, we relied on freely available precipitation and temperature data from WorldClim version 2 (available at http://worldclim.org/version2) (Fick & Hijmans, 2017) . These data consist of the 30-years average climate information (from 1970 to 2000) for the month of June, covering the entire world.

In an era of global change, an important feature of our method is that it is able to detect relatively subtle temporal changes in species composition. More specifically, we assessed whether group ranges had expanded north and contracted south. These are the patterns we a priori expected given warming temperatures and the strong influence of temperature on the spatial distribution of a range of taxonomic groups, including birds (Chen, Hill, Ohlemuller, Roy, & Thomas, 2011; Hitch & Leberg, 2007; Moritz et al., 2008; Parmesan & Yohe, 2003) .

To detect these patterns, we fit the model once to data from both time periods (instead of fitting the model separately for each time period 

We set the maximum number of groups to 20 for our case study. To interpolate the estimates of the proportion of different groups to unsampled areas, we relied on the inverse distance weighted (IDW) algorithm implemented in the package "gstat" (Graler, Pebesma, & Heuvelink, 2016; Pebesma, 2004) . Interpolations were restricted to locations within one degree of a BBS route. Finally, our algorithm was programmed using a combination of C++ (through the Rcpp package; Eddelbuettel & Francois, 2011) and R code (R Core Team 2013). We provide a tutorial in the Supporting Information Appendix S2 for fitting this model.

Despite assuming the potential existence of a much higher number of groups (K = 20), our results reveal that the proposed model was generally able to estimate well the true number of groups (boxplots in Figure 1) , except for datasets with few species and locations but many groups (i.e., 100 locations, 50 species, and 10 groups; Figure 1f) . We also find a good correspondence between the true and estimated parameter values for most of the scenarios explored (scatterplots in Figure 1) , with a slightly worse performance for data with few species but many groups (i.e., 50 species and 10 groups; Figure 1g,h) . Taken together, these results suggest that, when the ratio of the number of species to the number of groups is small, there is likely to be less distinction between groups from a species composition perspective, making it a harder task to untangle these groups.

Finally, in relation to the prior, we find that our results are broadly similar for γ ¼ 0:1 and γ ¼ 1. The main difference is that parameter estimates tended to be slightly worse when the true number of groups is 3 and γ ¼ 1 and when the true number of groups is 10 and γ ¼ 0:1 (results not shown), agreeing with our expectations.

Because smaller γ values induce greater sparseness, parameters are better estimated with γ ¼ 0:1 when simulations are based on sparse assumptions (i.e., simulations with three groups) versus when this is not true (i.e., simulations with 10 groups).

Our results reveal that the algorithm accurately estimates the proportion of the different groups in each location, regardless if MM units are present or not (left most and right most panels, respectively, in Figure 2 ). These results corroborate the observation that LDA encompasses hard clustering of sites and/or species as special cases. On the other hand, Figure 2 clearly reveals that hard clustering methods cannot represent these MM locations (k-means and hierarchical clustering [HC] panels in Figure 2 ). Mixture model approaches such as RCP are sometimes perceived to be able to represent these gradual changes in the proportion of groups. However, F I G U R E 1 The Latent Dirichlet Allocation (LDA) estimates well the true number of groups (boxplots) and the θ lk parameter values (scatterplots). Results from all 10 datasets in each simulation setting are displayed simultaneously, based on LDA with γ set to 0.1. Top and bottom panels display results for three and 10 groups, respectively. Boxplots in panels (a) and (f) show the estimated number of groups (i.e., the number of groups deemed not to be superfluous), revealing that LDA can estimate well the true number of groups (K*) except for datasets with few locations (L), few species (S) but many groups (i.e., 100L 50S 10K*). Scatterplots (panels b-e and g-j) reveal that the θ lk parameters can also be well estimated but there is considerable noise for datasets with few species but many groups (panels g and h). A 1:1 line and a linear regression line were added for reference (blue and red lines, respectively) [Colour figure can be viewed at wileyonlinelibrary.com] our results reveal that, when applied to our simulated data, RCP tended to give transition regions that were too narrow (RCP panels in Figure 2 ). These model comparison results are particularly striking given that LDA was fitted assuming 20 potential groups, whereas results for the other methods were based on the assumption that the true number of groups was known. Notice that these figures illustrate how LDA can capture gradual changes in species composition associated with global change phenomena depending on what is being represented in the x-axis. For instance, the x-axis can represent a spatial gradient of anthropogenic forest disturbance (e.g., timber logging intensity or distance to forest edge) or can represent time (i.e., the same location sampled repeatedly through time, perhaps revealing the impact of climate change on species composition).

Recall that the simulated data were generated with 3, 5, and 7 groups, but that the maximum number of groups when fitting LDA was set to 20. Our results suggest that the truncated stick-breaking prior was able to correctly estimate the underlying true number of groups (boxplots in Figure 3 ) given that the estimated θ lk 's were shrunk toward zero for the superfluous groups (red boxes in Figure 3) . We also find that all the other alternative methods required a much greater number of groups to fit the data as well as LDA when MM locations are present (line graphs in Figure 3 ). These results reveal that LDA achieves a much sparser representation of the data (based on the number of groups) without losing the ability to represent the inherent variability in the data. Although these results are expected, given the larger number of parameters in LDA, the ability to fit the data well with fewer groups is highly desirable from the user's perspective as the primary role of these methods is to reduce the dimensionality of biodiversity data. It is important to note that even in the absence of MM sampling units, LDA can still estimate well the true number of groups and has similar fit to the data as the other clustering approaches (results not shown). Finally, although 

Overall, we identified 18 main breeding bird groups (of a maximum of 20) after eliminating groups that were very uncommon throughout the region (defined here as those for which θ lk was smaller than 0.5 for 99% of the locations, where θ lk denotes the posterior mean).

An important test for any unsupervised method is if it is able to retrieve patterns that are widely acknowledged to exist by experts.

Using the estimated group proportion for each location for the 2010-2015 period, we find striking spatial patterns (maps in Figure 4) . Importantly, these spatial patterns generally agree well with other maps of bird communities (e.g., Bird Conservation (Figure 5a ). We find that the species that best F I G U R E 3 The extended Latent Dirichlet Allocation (LDA) method identifies the true number of groups (left panels) and fits the data better than other clustering approaches for data with MM locations (right panels). Results are shown separately for simulated data with 3, 5, and 7 groups (top to bottom). Boxplots depict the estimated proportion θ lk of each group k for all locations l = 1,…,L. These boxplots emphasize how θ lk for the irrelevant extra groups (red boxes) are shrunk to zero for all locations. Line graphs show the log likelihood, a measure of model fit for which larger values indicate better fit. These graphs reveal how other clustering approaches require a much greater number of groups to fit the data as well as LDA with fewer groups. Model fit results for LDA correspond to the posterior mean of the log likelihood. LDA results are shown with a single symbol because, differently from the other methods that were fitted multiple times with different number of groups, LDA was fitted just once using a maximum of 20 groups and the true number of groups was estimated (see corresponding boxplots). Details regarding how the log likelihood was calculated for the different methods are provided in Supporting find that group 10 identifies species associated with desert environments (e.g., cactus wren and ash-throated flycatcher), while group 8 identifies a mixture of short-grass prairie birds (e.g., dickcissel) and

species associated with open country environments with scattered trees and shrubs (e.g., eastern phoebe).

Besides these biogeographical patterns, we also highlight the ability of our algorithm in depicting how environmental gradients are linked to the proportion of each group. For instance, we display how the main East Coast groups (groups 2, 9, 12, 14, 15, and 19) are , 9, 12, 14, 15, and 19 along the East coast. (c) displays the spatial pattern of groups 8 and 10. In both (a) and (c), higher proportion of individual groups is depicted using more opaque (i.e., less transparent) colors and different groups are depicted with different colors. (b, d) reveals that average June temperature and precipitation gradients seem to strongly constrain the spatial distribution of these breeding bird groups, respectively. Circles represent the estimated proportion for each location and group while lines depict suitability envelopes. These envelopes were created by first defining equally spaced intervals on the x-axis and then calculating the median x value and the 99% percentile of y within each interval and connecting these results. Notice that the same color scheme is used for right and left panels

The Latent Dirichlet Allocation (LDA) model is a useful model for ecologists because it can more faithfully represent community dynamics and the impact of environmental change through the estimation of mixed-membership sites (Valle et al., 2014) . The standard LDA requires abundance data but, for many taxa, reliably estimating abundance is often very hard and costly (Ashelford, Chuzhanova, Fry, Jones, & Weightman, 2006; Joseph et al., 2006; Kembel, Wu, Eisen, & Green, 2012; Royle, 2004; Schloss, Gevers, & Westcott, 2011) . For these reasons, presence/absence data are typically much more ubiquitous than abundance data, often enabling analysis at F I G U R E 6 Species groups with a statistically significant association between latitude and change in group proportion Using the Breeding Bird Survey (BBS) dataset as a case study, we have shown how our method is able to uncover striking spatial and temporal patterns in bird groups. For example, we illustrate how these groups gradually change along a temperature gradient in the East Coast and a precipitation gradient in Texas. It has long been known that many bird species have strong relationships with abiotic gradients (Bowen, 1933) , but how these gradients can explain entire groups of species has remained elusive. Furthermore, we find subtle but pervasive changes in bird group proportions, changes which follow the expected patterns based on climate change (e.g., Parmesan & Yohe, 2003) . Half of the species groups (nine of 18) have expanded their northern range and shrunken their southern range.

This pattern is consistent with species-specific models of changes in bird distribution with climate change in the United States (e.g., Hitch & Leberg, 2007; La Sorte & Thompson, 2007) . Our results expand on these findings by illustrating how entire groups are shifting their spatial distribution. Nevertheless, a more formal test that accounts for the multiple factors that influence the spatial distribution of birds will be required to ultimately confirm whether climate change is driving the spatial distribution shifts that we have detected.

An important limitation of the method that we have presented is that the identified groups do not change over time, even though their spatial distribution may vary. In other words, θ lk may change with time but ϕ ks does not. This is particularly relevant in the context of climate change, where it is possible that the species composition of the groups themselves might be changing (Lurgi, Lopez, & Montoya, 2012; Stralberg et al., 2009; Urban et al., 2016) . Another important limitation in this study is that the proposed model does not take into account imperfect detection, a pervasive issue for wildlife sampling (MacKenzie et al., 2002; Royle, 2004) . This shortcoming can be partially attributed to inherent limitations in the BBS dataset, given that the estimation of detection probabilities requires very specific data types (e.g., repeated visits in occupancy models). It is also critical to highlight the importance of repeated observations per location given the relatively low information content in binary presence/absence data. Determining all the parameters in the proposed model, including the optimal number of groups, can be challenging in the absence of these repeated observations. Finally, although important broad-scale patterns can be identified and novel insights gained from post hoc analysis of LDA model parameters, as illustrated with our case study, these results rely on a two-stage analysis that does not take into uncertainty in the estimated parameters. Our ongoing work is focused on extending LDA to accommodate covariates through regression models built-in to LDA so that uncertainty can be coherently propagated when performing more formal statistical tests and when making spatial and temporal predictions.

Community ecologists have traditionally relied on fitting clustering models with different numbers of clusters and choosing the optimal number of clusters using metrics such as AIC and BIC (Fraley & Raftery, 2007; Xu & Wunsch, 2005) . Using simulated data, we have shown how the truncated stick-breaking prior can aid the determination of the true number of groups. We acknowledge, however, that the modeler still has to specify the hyperparameter γ and the maximum number of groups K. Using simulated data, we have found that setting γ to 0.1 often works well and that our model often identifies K groups if the true number of groups is equal or larger than K.

While this may be seen as an indication that K has to be increased when using real data, an extremely large number of groups defeats the purpose of dimension reduction, making it increasingly harder to visualize and interpret model outputs. Ultimately, we believe that the decision regarding the maximum number of groups K is a balance between what the data suggest and pragmatic considerations regarding how the results will be displayed and interpreted.

Our empirical example focused on large-scale biogeographical patterns. Nevertheless, this method could also be applied in a landscape-scale context, identifying spatial variation in community structure within general habitat types and across patches, or to analyze long-term temporal changes in time-series data of species composition (e.g., Christensen, Harris, & Ernest, 2018) . Given the ubiquity of presence/absence data in community ecology, we believe that the extension of the Latent Dirichlet Allocation model developed here will see a much wider use, becoming an important addition to the toolkit of community ecologists.

We thank the numerous comments provided by Ben Baiser, Daijiang 

http://orcid.org/0000-0002-9830-8876

Robert J. Fletcher Jr. http://orcid.org/0000-0003-1717-5707

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