Introduction

Degradation of ecosystems from past land use is a widespread phenomenon and a common target of ecological restoration and adaptive management [1], [2]. A key step in the adaptive management process is monitoring to evaluate progress toward objectives [2], where, for example, trend in a population of an appropriately chosen sensitive species may indicate progress. However, this raises the question of how best to approach the problem of trend detection in such actively managed systems. Managed landscapes often exhibit substantial influence from past land use, and a modeling strategy is required that must be at once flexible and sophisticated enough to handle the multiple sources of variation and uncertainty that arise in such settings. Modeling of trend in environmental monitoring has often been approached with generalized linear mixed models and maximum likelihood or restricted maximum likelihood parameter estimation techniques [3], [4]. However this becomes intractable with complex error structures and “random effects” terms in mixed-models often have no explicit interpretation or meaning, only vaguely informing questions about error-generating processes [5]. Bayesian hierarchical modeling offers a more explicit conceptual and technical framework for tackling many of the complexities likely to be encountered when evaluating trend in managed landscapes, and ecologists are increasingly turning to Bayesian hierarchical approaches [e.g. 5]. Despite the sometimes intractable computational demands, Bayesian hierarchical models generally provide more efficient and realistic accounting of uncertainty in parameter estimates through the specification of error terms at each level of the model, including the covariance parameters themselves [5], [6]. A clear-eyed assessment of uncertainty is a crucial element in the adaptive management process.

Spatial structure is one source of uncertainty that is particularly relevant to modeling population trends in actively managed environments. By definition, these settings inherit past land use patterns which can result in complex spatial structure. Environmental gradients may occur across the study domain, providing an additional exogenous source of spatial complexity. Endogenous sources of spatial structure such as dispersal in the target population may also occur [7] and both sources create a modeling environment rich with spatial information. Under this paradigm, spatial structure is more than just a violation of the independence assumption to be dealt with, but rather becomes a source of important ecological insight that can strongly influence observed trends [8], [9]. Hierarchical modeling offers increased opportunities for harnessing this spatial information that is typically ignored or lost in pooled error estimates obtained from classical single-level models [5], [6], [10], [11].

The results of hierarchical spatial models will therefore be of tremendous interest to land managers and other decision-makers because they can provide explicit estimates of trend from subpopulations such as management units or those defined by other relevant grouping factors. Estimates of trend pooled over multiple subpopulations may be misleading if trends are asynchronous; a pooled estimate can indicate positive trend even while some subpopulations are actually in decline [11]. Spatially heterogeneous land use is an example of how such a pattern might arise. Historical contingency is not easily specified in a single-level model, particularly when the details of past land use are poorly understood. However, because variation in trend caused by past land use is likely to occur in discrete areal blocks (e.g. among old fields), the hierarchical model offers a solution, by way of a grouping factor or stratification within which subsampling occurs.

This approach also establishes a means for dealing with anisotropy (non-stationarity) which also arises in such situations [5], [7], [12]. In a manner akin to stratified kriging [7], spatial covariance parameters can be allowed to vary among subpopulations easily in a Bayesian hierarchical framework. This is an important and accessible improvement over geostatistical models that incorrectly assume an isotropic spatial process, particularly for managed landscapes where boundaries among subpopulations are discrete.

We constructed a suite of Bayesian hierarchical models, each with increasing specification of spatial structure, to evaluate trend from annual samples of density counts in a spatially-structured population of the facultative wetland forb, camas lily (Camassia quamash [Pursh] Greene [Agavaceae]). Our study was conducted during 2005–2010, prior to and immediately following the cessation of grazing and mowing on the study area. We also evaluated trend in camas density from a reference site monitored during the same time period. Our restoration study site was bisected by roads and drainage ditches, resulting in distinct subpopulations with different land use histories. We harnessed information from this source of spatial structure by fitting subpopulation-specific intercepts and slopes. We modeled anisotropy with subpopulation-specific spatial covariance parameters. We considered our reference site to be quasi-pristine, having been managed as a protected area for 5 decades. Our objective was to determine if a positive trend in camas density was present following the initiation of passive restoration after accounting for key sources of spatial structure; we hypothesized it would be. Our fully-specified spatial model represents a hierarchical extension of the universal kriging model used in geostatistics to predict values at unobserved spatial locations [6], [10], although our immediate goal was to estimate structural parameters of the model and gain insight into population trend rather than to make predictions per se. Our approach merges two important developments in ecological modeling, geostatistics and Bayesian hierarchical modeling, and represents an application that should translate widely to other actively managed ecosystems with richly structured spatial domains.

Materials and Methods

Study system

Camas is a facultative wetland forb species associated with seasonally-inundated wet prairies of northwestern USA and southwestern Canada [13], [14]. It reproduces from large heavy seeds and from bulb offsets that often results in patchily distributed but densely populated colonies (Figure 1). The species was highly prized by indigenous people as a food source [13], [14], [15], and was the focus of major historical events that occurred during harvest that are today commemorated in two US National Park Service units. The extent of these wet prairie ecosystems has been drastically reduced in the region as a result of agricultural conversion and other land use practices [16], [17], [18]. Remaining wet prairies in the region are often structurally altered and compromised by herbaceous non-native and woody native invasive species, and some have been targeted for ecological restoration [e.g. 17,19].

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Figure 1. A photograph of camas growing in the restoration study site, Weippe Prairie, Idaho, USA.

Camas is a bulb-forming geophyte (upper inset) with limited dispersal capabilities. This attribute and its propensity for prairie swales that experience longer periods of inundation and elevated soil moisture during the growing season creates dense but patchily distributed colonies, as illustrated here. This patchy distribution also leads to highly skewed density counts (lower inset). Upper inset illustration by Andrea Foust Carlson, reproduced with permission. Photo courtesy of the National Park Service.

https://doi.org/10.1371/journal.pone.0028635.g001

In 2005 the National Park Service initiated camas monitoring in a 100-ha portion of the Weippe Prairie (hereafter, the “restoration site”), located in northern Idaho, USA (Figure 2; [20]). The site was acquired from private agriculturalists in 2003 to form a new subunit of Nez Perce National Historical Park. The site had been heavily used as pasture for livestock and some portions were regularly mowed for hay production. Passive restoration began immediately in 2003 with tapered grazing and mowing that was completely discontinued by 2008. Several proposed active restoration strategies were under consideration by park managers, including filling of drainage ditches to restore subsurface hydrology, an important motivation for our study. In 2006, monitoring was also initiated in a 20-ha portion of the Big Hole National Battlefield (hereafter, the “reference site”), located in southwestern Montana, USA [20]. This site has been under National Park Service management since 1963, a 5-decade period of effective conservation protection.

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Figure 2. The restoration site, Weippe Prairie, Idaho, USA.

The stratified sampling frame identifies 5 zones labeled A–E. A narrow ditch between zone A and B overlaps with the sampling frame boundary and is not readily visible. A paved road separates zone C and D. Elevations were obtained from high-resolution laser altimetry (LiDAR).

https://doi.org/10.1371/journal.pone.0028635.g002

Historic ownership and land use patterns in the restoration site followed the township-range-section pattern established during the US public lands surveys of the 19^th century, and are reflected in the contemporary pattern of roads and drainage ditches that partition the site (Figure 2). A straightened and deeply-incised stream channel bisects the study site. These features are substantial and impede surface and subsurface hydrology. Given the dispersal limitations of the species [14], we considered zones to be functionally disconnected by these historic landscape features. The monitoring program was designed to reflect this spatial structure with a stratified sampling frame [20], ensuring that adequate sampling effort occurred in each of 5 recognized subpopulations (hereafter, “zones”; Figure 2). In Figure 2, zones C and D were under different ownership than zones A, B and E prior to NPS acquisition. The intensity and type of land use is believed to have differed among owners, with hay production emphasized over grazing in zone D. Unfortunately, additional details of land use history are lacking for the site, although the condition of zone E was apparently very poor at the time of NPS acquisition and clear evidence of overgrazing in that area persisted until 2008 (Jason Lyon, National Park Service, personal communication).

Model building

We hypothesized that cessation of grazing and mowing, a chronic removal of above-ground photosynthesizing tissue of camas plants, would allow surviving plants to replenish carbohydrate stores and allocate more energy into reproduction, yielding an overall positive trend in camas density over time. However, we also expected that trends might vary among zones due to differences in the timing and intensity of historic uses of those zones. For example, zone D was mowed rather than grazed prior to NPS acquisition; such differences might influence contemporary patterns of density (the intercept) and rates of change in density (slope) among zones. Furthermore, given the association of the species with seasonally-inundated wetlands, we expected that prairie topography and the resulting spatial pattern in duration of inundation and soil moisture would also influence patterns of density within zones. Finally, given the limited dispersal capabilities of the species, we expected a strong pattern of residual spatial autocorrelation among observations, even after accounting for the influences of topographic heterogeneity.

We constructed a hierarchical spatial model using a conceptual framework and notational scheme following Wikle [6]. This approach involves the decomposition of a complex joint probability distribution with many parameters into a series of conditional distribution models representing the data-generating process, the latent spatial process, and the parameters. To allow for zonal variation, we indexed the spatial process and the parameter vectors by k, representing the zones, an additional level in the hierarchy. Letting represent camas counts observed in zone k over years 2005–2010, our fully spatial model is: (1)(2)(3)

where X = [1,year,elevation], β_k = [β_intercept,k, β_year,k, β_elevation,k]^T, µ_β = [µ_intercept, µ_year, µ_elevation]^T, and θ_k = {, φ_k}, the partial sill and range parameter [6], [10]. We used a lognormal model for Y to account for the rather extreme overdispersion observed in the study populations, in which plot counts were distributed with many zeros and very long positive tails of high density counts (Figure 1). We evaluated negative binomial and zero-inflated negative binomial distributional models to describe camas density via goodness-of-fit tests and exploratory models, but found that the high density counts were best described by the lognormal model. We accounted for spatial correlation among plot observations that shared the same (zonal) history by specifying zone-specific slopes and intercepts (equation 3) with common hyperparameters µ_β and [11]. We assumed an exponential covariance model for , where D_k is the Euclidean distance matrix for observations in zone k, and is the nugget of residual variance. We estimated the effective range, ξ_k = to make inferences about the practical extent of residual spatial autocorrelation [10], [22]. By indexing the spatial covariance parameters by k, our model allowed for anisotropy among zones, but enforced stationarity over time, an assumption supported by exploratory semivariograms shown in Figure 3a and 3b.

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Figure 3. Semivariograms of camas density in the restoration site.

In panels a and b, parametric exponential semivariogram models were fit to a) pooled and year-specific and b) pooled and zone-specific empirical models from model 3 residuals for the restoration site. In panel c, zone-specific posterior median estimates for partial sill and range covariance parameters (, φ_k) obtained from model 4 were used.

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We chose uniform(0,100) prior distributions for and normal priors on [11]. We used inverse gamma(0.1, 0.1) prior distributions for and following Banerjee et al. [10]. We used uniform priors for φ_k with informative lower and upper bounds. Following an approach described by Wang and Wall [23], we estimated the lower bound, φ_min = −log(0.5)×, and the upper bound φ_max = −log(0.01)×, where d_max and d_min were the maximum and minimum distances between plot observations. Diffuse priors for φ are problematic because it is typically a poorly identified parameter and a variety of approaches to prior specification have been suggested [e.g. 10,23]. The constraints imposed on φ_k allowed for a maximum correlation of 0.5 at the maximum distance between plots, and a minimum correlation of 0.01 at the minimum distance between plots.

In order to better understand how spatial structure influenced the estimation of trend, we also considered three reduced models: a hierarchical model with zone-specific slopes and intercepts but with no assumed residual correlation;

(4)

a partial-spatial hierarchical model as described by equation 4 but where X = [1,year] and β_k = [β_intercept,k, β_year,k]^T, excluding the elevation covariate; and a naive model involving separate regressions (“no pooling” sensu [11]) of camas density against year for each zone. For brevity, we refer to these as “model 4”, “model 3”, “model 2”, and “model 1”, respectively. Given our a priori assumption that the full spatial model would provide the most information and the most “honest” accounting of uncertainty, we based all inferences on model 4.

We fit models using OpenBUGS software [24] run from a multi-core processor computing platform with a Linux operating system (Text S2). This provided enough computational speed to obtain a sufficient number (40,000) of MCMC samples from the posterior distribution for model 4 which involved high-dimension covariance matrix decompositions. Following an initial burn-in period of 1000 samples, we thinned the subsequent 40,000 by a factor of 10, which was determined to be adequate from preliminary runs and evaluation of auto-correlation and convergence diagnostics. Inferences therefore were made on posterior distributions from the 3 chains each with 4000 MCMC samples. We evaluated convergence with the Gelman-Rubin diagnostic,; convergence was reached for all parameters according to the criteria [11]. Bayesian P-values were estimated from the discrepancy in the sum of squared residuals between observed and replicated data (i.e. posterior predictions) as a measure of model fit [11], [25]; P-values ranged from 0.50–0.54 for all models, indicating that the model adequately described the data. Empirical semivariograms shown in Figures 3a and 3b were estimated with the Hawkins-Cressie robust estimator [26]. We centered year and standardized elevation data, which made the intercepts more interpretable and improved parameter estimation and MCMC convergence [11]. We also centered spatial coordinates of the plot locations themselves following Banerjee et al. [10].

Results

There was clear evidence that camas density patterns varied strongly by zone. Point estimates for zone-specific intercepts, β_intercept,k, obtained from our full spatial model (model 4) ranged from −0.06 to 3.02, detailed in Table 1. Because camas density was modeled on the log scale and the year and elevation variables were centered, these intercepts represent zone medians of camas density (per 0.6 m² plot on the log scale) at mean elevation (916.8 m) in the middle of the study period. Zones B and E had the lowest overall density counts throughout the study period, averaging 8 and 5 plants m⁻², respectively (SDs = 18 and 10 plants m⁻²). Zone C exhibited moderate levels of density, averaging 28 plants m⁻² (SD = 45), although model 4 intercept variability (SD) was high (Table 1), as a result of a wide range of high and low density patches present in the zone. Zones A and D had the highest observed density counts, averaging 65 and 45 plants m⁻², respectively (SDs = 65 and 77), with some patches exceeding 200 plants m⁻². Density estimates in the reference site resembled that of zone E in the restoration site, averaging 5 plants m⁻² (SD = 12).

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Table 1. Parameter estimates based on 12,000 MCMC samples from posterior densities obtained from a fully spatial hierarchical model (model 4) fit to camas lily monitoring data collected in Weippe Prairie, Idaho, USA and to a reference site in Big Hole National Battlefield, Montana, USA, during 2005–2010.

https://doi.org/10.1371/journal.pone.0028635.t001

Trend estimates obtained from model 4 were positive for each zone in the restoration site as well as for the site overall, suggesting that passive restoration may be having a desired effect on the camas population (Figure 4a). However, there was considerable variation in the strength of evidence for trend among zones. Zone C exhibited the strongest trend over the 6-year period, with an estimated 16% (e^0.15) average annual rate of increase in median camas density, and a 95% credible interval (CI) around that point estimate of 0.09–0.22 (Table 1). However, the CI for zones A and B were also >0. There was evidence for a weak trend in Zone E with the lower CI = 0 (Fig. 4a). There was no clear evidence for trend in zone D (95% CI: −0.05–0.12) nor in the reference site during the same time period (95% CI: −0.02–0.07%; Table 1). The SD was consistently low (≈0.3) for trend parameters in all zones (Table 1), even lower for the reference site (0.2), and convergence was quickly reached in MCMC chains, adding confidence to our trend evaluation.  = 1.0 for all 5 zone trend parameters, as well as for the reference site trend parameter.

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Figure 4. Posterior medians and 95% Bayesian credible intervals for a) trend and b) elevation parameters.

Results are for the restoration study site from a series of 4 models of camas density each with successive amounts of spatial structure included. Model 1 represents a spatially “naïve” approach with estimates obtained from independent simple regression models for each zone. Note that no overall restoration site estimate (i.e. µ_year) is generated from model 1. Model 2 is a hierarchical model that allows for zone-specific slopes and intercepts as well as an estimate of µ_year for the restoration site (labeled “site”). Model 3 is an extension of model 2 with elevation included. Model 4 adds further spatial structure by including zone-specific exponential spatial covariance models that account for residual spatial autocorrelation.

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Topographic heterogeneity appeared to strongly influence patterns of camas density, particularly for zones C, D and E, which had the highest topographic relief on the site (Figure 2). The CI width for elevation was narrower for these three zones than for zones A and B (Figure 4b). As expected, camas density was negatively associated with elevation, and median camas density was estimated to decrease by ≈50–57% (exponentiated), depending on the zone, for a 1 SD (≈1 m) increase in elevation above the mean (Table 1; Figure 4b).

Estimates for the effective range (ξ) and partial sill (σ²) both varied among zones (Table 1; Figure 3c), supporting our expectations formed during exploratory analysis (Fig. 3a and 3b) for anisotropy and the need for separate covariance models for each zone. Semivariograms in Figure 3 illustrated that Zones A, C and D had the strongest residual spatial autocorrelation, which was much better described by an exponentially-decaying spatial covariance model than for zones B and E. The lower 2.5% posterior credible intervals for ξ from zones A, C and D ranged from 82–250 m (Table 1), providing evidence for dispersal-driven patchiness at a scale of 10's of meters, despite the high uncertainty in the exact location of those effective ranges (Table 1). Estimated correlation at 10 m lag distance was ≥64% for zones A, C and D, but only 34% for zone B, and <0.01% for zone E. Similarly, median ξ for the reference site was 78 m (Table 1), estimated with relatively high precision (95% CI 53–138 m), and a correlation of 40% at 10 m lag. The CIs for ξ_k in the restoration site were generally wide, but narrowest for zones A and D. Results for zone E indicated a complete absence of residual spatial autocorrelation (Table 1; Figure 3c).

Adding increasing amounts of spatial structure to the model, as represented by a progression from model 1 to model 4, influenced CI width and also adjusted the point estimates of trend (Figure 4a). Statistical “shrinkage” of trend estimates toward µ_year occurred, particularly in models 2 and 3, with a slight adjustment away from µ_year after accounting for residual correlation in model 4. A substantial increase in precision of trend (i.e. narrower CIs) was observed with models 2 and 3 after accounting for the exogenous sources of spatial structure, zonal correlation and topographic heterogeneity.

Finally, we note that topographic heterogeneity was very influential to both the point estimates and precision of trend in all zones, and was apparently sensitive to the meso-scale (e.g. 10–100 m) dispersal-related spatial autocorrelation in complex ways. A shrinkage effect stronger than that observed with trend occurred for elevation parameters in model 4, which shifted away from 0 toward µ_elevation (Figure 4b). There was a complex shift in precision among the elevation parameters as well, where in zones A, C and D, strong residual autocorrelation was apparent and the precision of the elevation parameter decreased in model 4 (Figure 4b). The opposite pattern was observed in zones B and E, where residual autocorrelation was weak.

Discussion

We developed a spatially-explicit hierarchical model to evaluate trend in the colonial geophyte, camas, following cessation of grazing and mowing in a wet prairie ecosystem. We also applied a simpler spatial model to estimate trend from a reference site for comparison. We found evidence for positive trend following passive restoration. Accounting for spatial structure made a substantial change in our evaluation of trend, however. Results from model 1, a set of independent spatially-naive models for each zone, resulted in rather equivocal evidence for trend; all zones except C included 0 in posterior densities (Figure 4a). Evidence strengthened as we incorporated progressively greater amounts of spatial structure. Results from model 4, our fully spatial model, provided compelling evidence for positive trend over the 6 years of study in 3 zones, and weak evidence in a 4^th zone. By comparison, we found no evidence for trend in the reference site over the same period of time, further supporting our conclusion that passive restoration seems to be having a desired effect in at least some portions of the restoration site.

The importance of accounting for underlying differences in zone-specific camas subpopulations was clearly evident. Intercepts, trend, elevation, and spatial covariance parameters all varied in complex ways among zones. High density zones with large intercepts appeared to have the strongest pattern of residual spatial autocorrelation. Zone E, an area that we believe was most heavily grazed prior to and during the first 3 years of monitoring, had very low camas density and no residual autocorrelation. In Zone B, another portion of the restoration site that appeared to exhibit lower than expected density, residual autocorrelation was only weakly evident (Figure 3c). In contrast, the reference site, which had a similar level of density to zones B and E, exhibited the most precise signal for spatial autocorrelation out to a lag distance of ≈80 m. This is interesting because it suggests that colony development and the resulting exponential covariance pattern may strengthen over time in the absence of chronic disturbance. Based on considering residual correlation at 10 m lag and the lower CI boundaries for the effective ranges across zones and in the reference site, a distinct pattern of dispersal-driven patchiness may be typical for our two study populations out to several decameters, but may weaken and become more variable under chronic disturbance and degradation. After accounting for topographic heterogeneity, the two low-density zones believed to be most altered by past land use exhibited the most homogeneity in patch structure. Camas, where present, typically occurred as single plants loosely assembled in undefined patches. This suggests that colony formation might become more distinct over time, reflected in increasing precision of covariance parameters.

The shifts in both the location and scale of trend posterior distributions were noteworthy. Accounting for exogenous spatial structure, i.e. zonal differences and topographic heterogeneity, resulted in increased precision in trend as variation was “mopped up”. Shrinkage effects resulting from the hierarchical construction of the model involving the use of hyperparameters for trend pulled posterior densities closer together and generally away from 0. This makes sense, even for zones with weak effect sizes – in the absence of strong information in one direction or the other, µ_year provides the best expectation for trend [11].

Less intuitive shifts in location and scale occurred with the elevation parameter that appeared to reflect a complex interaction with residual spatial autocorrelation. In model 4, there was a strong pattern of shrinkage away from 0 toward µ_elevation. This shrinkage effect was not seen in model 3, perhaps due to confounding effects of spatial autocorrelation. The scales at which the spatial processes of topographic heterogeneity and dispersal-driven autocorrelation occur in the restoration site are at distances of tens of meters and therefore overlap. Furthermore, dispersal likely follows the swales with elevated soil moisture in such a way that patch boundaries reflect the underlying topography (Figure 1). This is the likely cause for the divergence in CI widths among zone elevation posteriors seen in Figure 4b. Precision in the elevation CI increased from model 3 to model 4 in zones B and E, the two zones with weak spatial autocorrelation. Precision decreased predictably for zones C and D, and to a lesser extent for zone A, all 3 of which had much stronger residual autocorrelation.

The phenomenon of coefficient shift in regression models when spatial autocorrelation is present yet not correctly modeled has been a topic of recent debate [e.g. 27,28]. Conflicting reports have been made concerning the predictability of shifts in both location and scale in the covariates. In general it is well established that spatial structure in both predictor and response variables result in inflated Type I error [28], [29], [30], as was observed in our inflated precision of the elevation parameter for zones C and D, corrected in model 4. A similar correction was observed in zone D trend, as well as in µ_year. What was striking about our results, however, was that variation in those shifts among zones depended on the strength of autocorrelation and amount of topographic heterogeneity. Furthermore, the shrinkage effect induced from “partial pooling” of slopes and intercepts apparently can override expected coefficient shifts. Beale et al. [28] observed that covariate effect sizes are typically smaller in spatial regression models when residual spatial autocorrelation is present, particularly when true effect sizes are near 0. They suggested this is because non-spatial regression yields less precise estimates, allowing the magnitude of estimates from structurally incorrect models to vary widely. However, our results demonstrate another scenario: in hierarchical models that index covariates by grouping factors, parameters for groups with small effect sizes will shift toward the common mean rather than 0. Groups with small sample sizes will also shift toward the mean because they have less “information” to provide [11]. Modeling spatially-correlated errors therefore can result in a strengthening of that shift, as was demonstrated in Figure 4b, by lowering the effective sample size in groups with strong autocorrelation. We have not found this phenomenon described elsewhere and it was not clear from our study whether this result can be anticipated generally, but it is a scenario likely to be encountered more frequently as the use of hierarchical models increases among ecologists.

Our fully spatial model is a hierarchical extension of the geostatistical kriging models commonly employed for spatial prediction of natural resources [e.g. 10]. Isaaks and Srivastava [12] differentiated between geometric anisotropy, where ranges differ but sills are the same, and zonal anisotropy, where ranges are the same but sills differ, with each type requiring a different strategy to recover the necessary assumption of stationarity [7]. We encountered both types in our study and the flexibility of the Bayesian hierarchical approach allowed us to specify independent covariance models for each zone to accommodate this complex anisotropy. Our strategy is akin to stratified kriging used to interpolate across distinct soil types [31] and forest stands [32]. In situations where anisotropy is not so discrete, this approach may not be appropriate, but we envision many situations where disjunct “steps” in residual covariance patterns are likely, particularly in landscapes modified by human agriculture and roads. Explicitly modeling discrete patterns of spatial covariance improves interpretation and provides insight into spatial processes otherwise masked by globally isotropic models, as we have demonstrated. Furthermore, our approach reduced the n-dimensional covariance matrix to smaller n_k-dimensional matrices, a useful computational strategy when estimating spatial covariance parameters by way of complex matrix inversions in geostatistical models (i.e. the “big n problem”, [10]).

Supporting Information

Acknowledgments

Access to the University of Idaho Institute for Bioinformatics and Evolutionary Studies (IBEST) Unix computing core was facilitated by James Foster, Robert Lyon, and Zaid Abdo. The authors would like to thank the many National Park Service staff and student citizen scientists who assisted with data collection. Gerry Wright and Jason Lyon provided helpful comments during manuscript preparation. This is contribution 1063 of the University of Idaho Forest, Wildlife and Range Experiment Station. Any use of trade, product, or firm names is for descriptive purposes only and does not imply endorsement by the U.S. Government.