Introduction

Colonization by non-indigenous species is a leading environmental issue (Sala et al. 2000; Mooney et al. 2005; Strayer et al. 2006) and an important component of global change (Mooney and Hobbs 2000) and biotic homogenization (Rahel 2002). In North America, freshwater ecosystems support a large endemic fauna and provide valuable ecosystem services and are therefore particularly vulnerable to invasion (Vanderploeg et al. 2002). Thus, forecasting invasions in freshwater ecosystems has been identified as a leading challenge for ecological forecasting (Clark et al. 2001; Jones and Ricciardi 2005) and risk analysis (Bossenbroek et al. 2005).

Disentangling the relationships between species and their environments—species niches—is crucial for forecasting biological invasions (Peterson 2003). Though the niche is a fundamental ecological concept (Chase and Leibold 2003), niche identification has been plagued by conceptual ambiguity and technical obstacles (Pulliam 2000). Further, when identifying niches for invading species, the distribution of observations available for modeling is necessarily non-stationary, as the invading species is progressing across a landscape encountering new and different environments, violating the assumptions of most conventional statistical methods. The result is that niche models are commonly severely biased.

We used a machine-learning approach to overcome these obstacles and profile vulnerability to invasion by zebra mussels (Dreissena polymorpha) for 1,017 lakes and reservoirs in the Great Lakes region of North America. The zebra mussel is a nuisance in freshwater ecosystems in North America (Schloesser et al. 1996; Ricciardi et al. 1998) and Europe (Karatayev et al. 1997), dramatically altering ecosystem cycles (Strayer et al. 1999; Vanderploeg et al. 2002), fouling underwater industrial infrastructure (O’Neill 1997), and threatening the viability of endemic clam populations (Ricciardi et al. 1998). Zebra mussels have been designated by the World Conservation Union Invasive Species Specialist Group as one of the one hundred worst invasive species worldwide (http://www.issg.org/database). Here, we set up a framework for computational niche identification and show how the support vector data description (SVDD), a machine-learning algorithm for one-class classification, can be used for niche modeling. Following this, we deploy the approach to profile ecosystem vulnerability to invasion by zebra mussels across the Great Lakes region of the United States.

Niche theory

Hutchinson (1957) defined the ecological niche as the set of environments in which a species can persist. The implied subjunctive conditional—that if species S was introduced to environment E, and E was a member of the niche, then S would persist in E—has been a sticking point for niche identification, since data on persistence are typically available from only a fraction of possible environments. We represent the relevant features of a species’ environment at a location i by a vector of observations or measurements corresponding to each of n niche axes, \( {x_i = \left[ {y_{i,1}, y_{i,2}, y_{i,3}, \ldots, y_{i,n} } \right] \in E} \), where is the set of all (possible) environments (Hutchinson 1957; Chase and Leibold 2003). Through the use of appropriately specified niche axes, any quantitative or qualitative characteristic of the environment is allowed. Thus, a niche axis may take as its domain the real numbers, non-negative integers, or binary indication {0, 1}, and possibly others. The domain of the niche axis is the set, , of possible niche conditions y. Examples are temperature (a continuous quantitative variable), the number of competitive species in the community (a discrete quantitative variable), soil type (a categorical variable), and the presence of a mutualist symbiont (a binary indicator). While not a part of our analysis below, in general other species or combinations of other species might be considered as niche axes. Historically, this has been a point of confusion, i.e., in the theory of competitive exclusion where one conception of the problem uses niche restriction to detect competition while another conception holds that competition is constitutive of the niche, being part of its definition. We remark that these differences are matters of convention and that our conception of the niche can be modified to accommodate these distinctions and others by partitioning into subsets.

We now turn to the criterion of persistence. Roughly, populations persist when average individual fitness, λ, a function of the environment and population size, is greater than one, i.e., λ(x, z) > 1, for some population size z, z<<z max, where z max is the maximum size of the population. The niche may be therefore defined as . Finally, for any niche axis there will be a subset of habitable conditions, , the set of all conditions that appear in any environment x belonging to the niche, . In short, the only niche conditions which are not habitable are those that do not belong to any niche environment. Importantly, some regions of the niche may not occur in nature. Thus, following Jackson and Overpeck (2000) we define the realized environmental space, , as the subset of environments realized in nature and subsequently define the potential niche, the subset . Niche identification, the estimation of the extension of , from observations drawn from is generally a difficult problem.

Computational niche identification

The goal of identifying (or ) is not original to this study (Grinell 1917), and is a necessary step for numerous applications in ecology, evolutionary biology, and environmental science. We submit that our approach, described below, is optimal in the sense that it eliminates unwarranted theoretical restrictions (particularly relaxing the requirement that the niche include all environments in the product space of habitable conditions; cf. Hutchinson 1957; Stockwell 2007) while retaining the intuition that the niche should be connected. Connected, in this sense, means that any niche environment x is reachable from any other niche environment by a series of operations on the elements of x in which an element is individually incremented or decremented within the local neighborhood of nearest points. While niches need not be continuous (some niche axes may be discrete), it is an empirical conjecture that they are not disconnected in the sense that two niche environments may be separated by an intermediate non-niche environment. Equivalently, niche and non-niche environments may be separated by a hyperplane.

Ideally, niche identification would proceed by estimating λ(x, z) directly from observed data on individual growth, survival, and reproduction (cf. Pulliam 2000). Obtaining such data is costly, however, requiring intensive long-term fieldwork. Thus, one typically assumes that much less costly observations of species occurrence are indicators of local persistence and therefore a reliable surrogate for high fitness environments. In terms of our notation, numerous methods for identifying (but not generally ) based on a set of occurrence data and associated environmental measurements have been proposed in recent years (Hirzel et al. 2002; Elith et al. 2006; Pearce and Boyce 2006; Phillips et al. 2006). Some of these methods have shown remarkable accuracy when compared with validation data and reported accuracy has improved as increasingly sophisticated methods are introduced (Stockwell and Peterson 2002; Elith et al. 2006). However, all methods with which we are familiar fail to fully avoid one or both of the following problems. First, data are typically unbalanced with observations of species occurrence vastly outnumbering confirmed locations of species absence, if any absence data are available at all. Further, misclassification errors in the data are also asymmetrical (true occurrences are more likely to be classified incorrectly as absences than vice versa), either due to sampling error, or (as in the case of a species invasion) because the current true distribution of environments inhabited by the species is transient. The severity of this problem will vary among species. Clearly, for organisms that disperse slowly, experience frequent local extinction, or are unable to thoroughly explore pockets of habitat (perhaps due to fragmentation) this problem will be severe. These and other problems pertaining to the balance of observations have come to be known as the problem of “presence-only data” since the limit case comprising only observations of species presence with no observations of species absence is the most commonly available form of data (Hirzel et al. 2002; Brotons et al. 2004; Pearce and Boyce 2006). Second, data are generally not independent. Regardless of how data are obtained (i.e., by computationally sampling from maps of species distributions with GIS, merging records from museum collections, or new field collections), they represent geographically distributed populations in which autocorrelation will be high at some spatial scales (and may not be isotropic) and for which the grain of subsampling that would ensure statistical independence is unknown. Often data exhibit other unknown investigator-induced correlations as well. In our view, these problems—the balance of observations and non-independence—conspire to seriously confound hypothesis tests and estimation. The effects of these problems are exacerbated when sampling in not even across the species distribution.

Because of the need to avoid these problems, niche identification methodology is now an active area of research (Guisan and Thuiller 2005; Moisen et al. 2006). New techniques have been introduced to overcome these problems, but to our knowledge none addresses both. Heuristic methods to circumvent the problem of presence-only data include simulating species non-occurrences (Stockwell and Peters 1999; Engler et al. 2004) or avoiding the classification formulation altogether by estimating the multivariate distribution from which observations are drawn (e.g., Hirzel et al. 2002; Phillips et al. 2006). In the former case, there are well-known theoretical problems and some practical ones (Hirzel et al. 2002; Pearce and Boyce 2006), though the procedure has sometimes proved reliable (Anderson et al. 2003). In the latter case, the estimated distribution does not in general reflect the actual distribution of eventually colonized habitats because it is affected by the distribution of habitat availability, the sampling process, the spatial trajectory of the invading species, the process of resource selection by organisms, and the spatial configuration of different environments (Keating and Cherry 2004; Manly et al. 2004; Lele and Keim 2006). Often, the autocorrelation problem is ignored (for an example from our own research, see Drake and Bossenbroek 2004; cf. Wood and Augustin 2002; He et al. 2003).

We suggest an alternative approach to forecasting species invasions that proceeds in two steps. First, to identify niche axes, use a local probabilistic regression method that controls for effects due to spatial autocorrelation to identify variables that are important predictors of the species present transient distribution. We use geoadditive modeling, though for a population that has achieved a stationary spatial distribution autologistic regression might be preferable. Second, use statistically significant variables to estimate niche boundaries, i.e., contours of extreme values of the distribution from which the observations are drawn or a decision boundary for classification of niche from non-niche environments. Importantly, however, we advocate using methods that make limited assumptions about the statistical properties of actual observations. Given the complicated conditional relationships among the processes leading to the distribution of observations, a non-probabilistic approach could be particularly useful. We applied this two-part approach to a dataset on zebra mussel presence in 1,202 lakes and reservoirs across the Great Lakes region of the United States.

Materials and methods

Data

As for other mollusks, water chemistry is an important determinant of zebra mussel habitat. Specifically, dissolved calcium and pH are critical for larval survival, growth, and shell formation (Vinogradov et al. 1993). An earlier study found that these two variables predicted occurrence in Europe with 92.7% accuracy when data on both occurrence and absence were available (Ramcharan et al. 1992). Accordingly, we envision that each lake is represented by the pair x i  = [y 1, y 2], corresponding to pH and dissolved calcium concentration. Water quality data were obtained from the EPA STORET data bank (http://www.epa.gov/storet/index.html) and the USGS National Water Information System (http://waterdata.usgs.gov). From each data source, we retrieved records of pH and dissolved calcium for all lakes for which they were available in each of the eight states that contain multiple inland lakes or reservoirs infested by zebra mussels (Illinois, Indiana, Michigan, New York, Ohio, Pennsylvania, Vermont, and Wisconsin). Kansas and Oklahoma together have only three reservoirs with zebra mussels and were excluded from the study. Records of pH and calcium were attributed to individual lakes using a geographic information system (ArcGIS, ESRI, Redlands, California). To diminish the influence of spurious outliers, the top and bottom 5% of parameter values for each lake were trimmed before calculating mean dissolved calcium and pH for 185 invaded lakes and reservoirs and 1,017 non-invaded lakes and reservoirs. Dissolved calcium data were log10-transformed prior to analysis.

Hypothesis tests for niche axis identification

We tested for an association between zebra mussel occurrence and our two putative niche axes (dissolved calcium and pH) using geoadditive models (Kammann and Wand 2003). Geoadditive models are generalized additive models (GAMs) in which the two-dimensional effect of space is included as a covariate. For the reasons discussed above, we chose to use geoadditive models to accommodate the highly probable situation in which invasion status of lakes is spatially correlated. For zebra mussels the causes of spatial autocorrelation include proximity to propagule sources, density of habitable ecosystems, stream connections, geospatial variation in the human behaviors resulting in introductions, and the effect of spatially correlated covariates, particularly large-scale features of bedrock and surface geology. Our model is fully specified by the assumption that the binary response variable (invaded or not invaded) was binomially distributed with mean

$$ {\eta = t\left( {x,y} \right) + s\left( {z_0 } \right) + s\left( {z_1 } \right) + s\left( {z_2 } \right)}, $$

where t(x,y) is a tensor-product smooth interaction (with thin-plate regression spline basis) describing the spatial effect at longitude x and latitude y, and the s(z i) are thin-plate regression splines for effects of dissolved calcium, pH, and lake area (Wood 2006). Lake area was included as a measure of lake attractiveness (Reed-Andersen et al. 2000; Bossenbroek et al. 2001), which in turn affects inbound propagule pressure and could therefore be a confounding factor (Leung et al. 2006). Models were fit in the statistical programming language R (R Development Core Team 2007) with the mgcv package (Wood 2001); tuning parameters were selected automatically minimizing the unbiased risk estimator (UBRE), which can be interpreted as an approximation to Akaike’s Information Criterion for nonparametric models (Wood 2004).

Niche boundary identification and risk profiling

Support vector machines (SVMs) are a class of non-probabilistic statistical pattern recognition algorithms for estimating, among other quantities, the boundary of the set from which a collection of observations is drawn. By design, SVMs are insensitive to large numbers of similar observations, thereby circumventing the autocorrelation and nonstationarity problems. Operationally, SVMs make use of a function (the kernel) to project observations with complex statistical properties in the natural input space (e.g., the niche axes) into a higher dimensional feature space in which they are more simply represented, for instance for a classification problem by a separating hyperplane (Schölkopf and Smola 2002). For niche identification, the goal is to estimate a function that distinguishes from using only observations from —the observations of species occurrence. The function should return a set of points in the input space that represents a boundary between the two classes, while the assumptions about the data and the target class must be minimal. In statistical pattern recognition, the task of identifying counter examples to a set of training data (i.e., points belonging to ⌝) is referred to by the nearly synonymous terms “novelty detection”, “one-class classification”, and “concept learning” (Manevitz and Yousef 2001; Tax 2001; Markou and Singh 2003). SVMs for novelty detection satisfy these conditions. Particularly, it is not assumed that data are independent, that data are distributed in proportion to the true distribution of the species in the environment, that the niche is a convex set, or that the observations agree in expectation with the average of the true distribution, (cf. Phillips et al. 2006). We do assume that the niche is not a disconnected subset of , and optimize the tuning parameter to ensure that the set of environments comprising the estimated niche is simply connected. Finally, we remark that accuracy will improve with the degree to which the distribution of observations are representative of the true potential distribution, in the sense that the range of the distribution has been sampled even if in a biased and unknown fashion. Particularly, performance will be improved when the boundary of the unknown distribution actually sampled, the boundary of , is coincident with the boundary of . We believe that this is the minimally restrictive approach to representing the ecological niche as classically defined by Hutchinson (1957). To our knowledge, the similarity between the problem of novelty detection and niche identification has not been remarked previously.

The support vector data description (SVDD) is an SVM for finding the boundary around a set of observations (Tax and Duin 2004). This boundary is the simplest boundary in the sense that it represents the smallest possible hypervolume (a hypersphere) containing a specified fraction of the observations in the projected feature space. The SVDD retains the general property of being insensitive to the distribution of the training data as long as the data are representative of the set of possible observations (Tax 2001). In particular, extreme observations provide a great deal of information about the boundary of the niche which can be exploited by the SVDD, but (if over-representative of their actual occurrence) would give a biased estimate of the distribution of the species in nature. Indeed, by avoiding estimating the density directly, the SVDD obtains a better estimate on the boundary of the niche than would be obtained from estimating the full density and “backing out” its supporting set from contours of tail probabilities (Tax and Duin 2004). Further, consistent with Vapnik’s principle to avoid solving a more general problem as an intermediate solution to a particular objective, the SVDD seeks to extract maximal information about a particular feature of a distribution (its boundary) by avoiding estimating unnecessary features (e.g., central tendency, dispersion, and skew; see discussion in Tax (2001), pp 67-ff.). For niche identification, this should result in maximally accurate estimates of niche boundaries.

Model training and performance estimation

Prior to model estimation, occurrence data were randomly assigned to model training (80%) or model testing (20%) subsets. Using only the training dataset for model estimation, the SVDD was obtained by solving the quadratic programming problem

$$ {{ \min }\left\{ {\sum\limits_{i,j} {\alpha_i } \alpha_j K\left( {x_i x_j } \right):\sum\limits_i {\alpha_i } = 1,0 \le \alpha_i \le C} \right\}} $$
(1)

where K(x i,x j) is the kernel describing the similarity between vector-valued object x i and x j, the α’s are the Lagrange multipliers, K is the Gaussian radial basis function (RBF) \( {K\left( {x_i, x_j } \right) = \exp \left\{ { - {{\left\| {x_i - x_j } \right\|^2 } \mathord{\left/ {\vphantom {{\left\| {x_i - x_j } \right\|^2 } {s^2 }}} \right. } {s^2 }}} \right\}} \) with tuning parameter s, and C is a parameter that trades off the volume of the hypersphere against the errors (Tax 2001). Our rationale for choosing the Gaussian RBF kernel was twofold: (a) its theoretical underpinnings suggest this kernel can be understood as a generalization of, and compromise between, the rigid hypersphere and Parzen density estimators (Tax and Duin 2004), which alternately may be considered as the extreme cases of the boundary estimation approach adopted in this paper (the hypersphere) and the density estimation approach commonly adopted (Parzen density estimator), and (b) our experience is that it is the most numerically stable of the standard kernels.

For model estimation, we only used pH and dissolved calcium. We suspect that large lakes are invaded first, because of high rates of human visitation, and that the significant effect of lake area obtained in the geoadditive model is at least accentuated during the early phases of invasion and probably temporary altogether. The parameter C was determined implicitly by defining a tolerable error rate on the target distribution \( {\varepsilon \in \left\{ {0.1,\,0.05,\,0.025} \right\}} \). Because of the very high economic and environmental costs of zebra mussel invasion (Leung et al. 2002), we submit that ε = 2.5% (representing one in forty chance of misclassifying a niche ecosystem as non-niche) is the maximum tolerable error rate in practice and focus our subsequent discussion on these analyses. The selection of s was automated to obtain the simplest description of the target set subject to the specified error ε. For detailed formulas for the estimated niche boundary and the associated test procedure we refer the reader to section 2.1 of Tax and Duin (2004).

The false negative error rate was estimated by executing the trained algorithm on the 20% of occurrences in the testing dataset. Estimating the false positive rate is trickier. Since the invasion is ongoing, susceptible lakes not yet invaded and properly identified as susceptible would score as false positives, the estimated false positive rate obtained by the a posteriori classification of lakes known not to be invaded could be severely biased. Therefore, we adopted an alternative suggested by Tax and Duin (2002) and estimated the false positive rate as the fraction of the sphere with radius equal to the radius of the target data occupied by the trained classifier. We recognize that the distribution of non-niche ecosystems in nature may not reflect the uniform distribution generated using this procedure and that some generated outliers will belong to the true target distribution (so that the error rate estimated on the simulated outliers is overestimated). Tax and Duin (2002) suggest that this method will be most prone to failure when observations from the “target class” (i.e., niche environments, ) are “scattered over the complete feature space” (i.e., the space of possible environments, ). Equivalently, one can readily appreciate that the distribution of outliers generated this way will be most accurate when (a) the true boundary of set is a highly restricted subset of (so that few outliers are generated within the niche space), and (b) the set of realized environments is evenly distributed over the domain . Recognizing these rather restrictive limitations, in this paper, we primarily use the estimated false positive rate to compute the receiver–operator curve, to facilitate evaluating our procedure and comparing with other models. Given the positive bias in the estimated false positive rate, our estimate of the summary area-under-curve statistic (AUC) should be conservative.

All data processing and estimation was performed in MATLAB version 7.2 (Mathworks, Inc., Natick, Massachusetts). Solution of the SVDD program in Eq. 1 was performed using the Data Description Toolbox (version 1.5.4, [Tax 2006]) and Pattern Recognition Toolbox (version 4.0, [Duin et al. 2004]).

Results

Invasion extent

Zebra mussels were found in 185 of 1,202 lakes in our sample. Of these, five are in Illinois, two are in Indiana, 119 are in Michigan, 19 are in New York, 11 are in Ohio, five are in Vermont, and 25 are in Wisconsin. In a few instances, larger lakes are represented by more than one observation because parts of the lake belong to separate USGS polygons. Thus, as an extreme example, Lake Champlain is included in our data set seven times. Average measurements of the trimmed data series for dissolved calcium ranged from 0.83 mg L−1 to 586.43 mg L−1, while dissolved calcium in infested lakes was between 6.2 mg L−1 and 436.39 mg L−1 (median: 31.6). Average measurements of pH ranged from 3.9 to 10.3, while pH in infested lakes was between 6.9 and 9.0 (median: 7.8).

Niche axes

All effects in the geoadditive model were significant (p < 0.0001; R 2 = 0.37; Table 1). However, the estimated degrees of freedom (e.d.f.) for the effect of pH was indistinguishable from one, so a second model was fit in which pH was included as a linear covariate. All effects retained their significance in this second model (Table 2). The effect of dissolved calcium increased noticeably from low to intermediate values, at which it levels off at a threshold around 1.5 × 101 mg L−1 (Fig. 1). Interestingly, the estimated coefficient for the effect of pH is negative, i.e., conditioned on dissolved calcium, pH has an inhibitory effect on mussel establishment that is constant across the observed range. Exponentiation of the fit coefficient obtains an estimate of the effect of increasing pH by one unit on the odds of having been invaded by zebra mussel, i.e., e−0.66≈0.5 indicates that the odds are reduced by approximately half.

Table 1 Output for the first geoadditive model with all effects represented as smooth terms
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Table 2 Output for the final geoadditive model with linear effect of pH; all other effects represented as smooth terms
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Fig. 1
figure 1

The partial relationship between infestation by zebra mussels and the concentration of dissolved calcium (after log10 transformation) increases to a point and then levels off in a geoadditive model. Plot shows mean effect with 95% confidence intervals

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Invasion risk profiles

Out of 1,017 presently uninvaded lakes, 645 fall within the estimated zebra mussel niche at the most restrictive false negative error tolerance of ε = 0.025 (Fig. 2). Relatively few of these are excluded when the error tolerance is increased to ε = 0.08 (Fig. 3). The zebra mussel niche appears to be a relatively restricted subset of the environmental space occupied by lakes in general. Vulnerable lakes were identified in all states for which analysis was performed, but are concentrated in the Midwest, particularly in Michigan’s Lower Peninsula (Fig. 3). Individual lakes and their vulnerability status are listed in Appendix S1.

Fig. 2
figure 2

The support vector data description estimated niche for zebra mussel (Dreissena polymorpha) in North America. Circles (red) represent observations of zebra mussel occurrence and dots (grey) represent lakes for which zebra mussel has not been reported. The model was trained only on observations of occurrence. The blue line is a non-probabilistic estimate of the niche boundary tuned to contain 97.5% (ε = 0.025) of occurrence observations, the magenta line contains 95.0% (ε = 0.05) of occurrence observations, and the red line contains 90.0% (ε = 0.08) of occurrence observations

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Fig. 3
figure 3

Map of lakes predicted to be susceptible to zebra mussels. Susceptible (colored) or unsuitable (black) lakes are based on the ecological niche identified with a support vector machine. Lakes colored red occupy the most interior region of the niche (ε = 0.08), followed by lakes colored magenta (ε = 0.05) and lakes colored blue (ε = 0.025). A table listing the individual lakes is available as Appendix S1

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The single tuning parameter (s) was selected to obtain the simplest model consistent with the target error rate in cross-validation. It is our experience that model results are generally remarkably insensitive to the choice of this parameter. Figure 4 shows the cross-validation error rate as a function of the tuning parameter. Ideally, this figure would show a very steep shoulder with a sharp turn in the vicinity of the target error rate. In this case, a somewhat simpler model might have been chosen by tolerating an error rate twice as large, but the actual estimated niche boundary is not markedly complicated (cf. Fig. 2). That is, there are no irregular corrugations in the estimated boundary. Our confidence is further supported by inspecting the receiver–operator curve, which shows the variation in the true positive rate as ε is varied over its possible range. The estimated AUC was 0.88, while the receiver–operator curve is quite flat in the vicinity of the target error rate ε = 0.025 (Fig. 5).

Fig. 4
figure 4

Model complexity is determined by tuning the width parameter (s) to obtain the simplest model that reliably achieves an average error rate less than or equal to the target value \( {\varepsilon \in \left\{ {0.1,0.05,0.025} \right\}} \) in cross-validation. The arrow points to the value used for drawing the boundary in Fig. 2 and predicting lakes shown in Fig. 3 at the target error rate of ε = 2.5%

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Fig. 5
figure 5

The receiver–operator curve showing the rate at which truly susceptible lakes are correctly identified (true positive rate) as a function of tolerance for falsely classifying unsuitable lakes as susceptible (false positive rate). The circle shows the point on the curve at the target error (false negative) rate of 2.5%

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Discussion

Using data from publicly accessible databases, we profiled the risk of zebra mussel invasion for lakes and reservoirs of the Great Lakes region of the United States. Our method circumvents most problems associated with presence-only data and unknown correlations, including spatial autocorrelation. Our results confirm that dissolved calcium and pH within observed ranges are determinants of zebra mussel habitat. Further, we have identified individual ecosystems at risk for zebra mussel invasion (S1). Our analysis demonstrates that the ecological niche, conceptualized as a subspace in the set of possible environments, can be estimated using support vector machines, and that computational approaches to data mining large, heterogeneous ecological datasets can be marshaled for ecological forecasting. Long-term ecological data are commonly beset by unknown dependencies, unbalanced observations, and non-random sampling. Our study shows that non-probabilistic machine-learning approaches to data analysis can avoid some of these obstacles.

We confirmed that two hypothesized niche axes (pH and dissolved calcium) significantly affected invasion status of lakes in the United States, independent of possible confounding influences of spatial correlation and lake area. Interestingly, pH was found to have an unexpected inhibitory effect, though compared with other factors (dissolved calcium, lake area, and spatial location) the evidence for an effect of pH is weak (p = 0.035). A speculative explanation is that this effect results from reactions of the different carbon species as pH varies. As pH increases, the majority of dissolved carbon changes from free CO2 to bicarbonate (HCO −3 ) to carbonate (CO 2−3 ), affecting relative proportion of each (Wetzel 2001). Consequently, when pH is high, i.e., >8.5, calcium is primarily bound by carbonate and therefore unavailable for uptake by mussels. Alternatively, eutrophy is often correlated with pH (Jeppesen et al. 1990), and hypereutrophic lakes are known to be poor habitat for zebra mussels (Strayer 1991). However, given the relatively weak evidence for the association between pH and occurrence, future research should confirm that this pattern is not spurious. After identifying pH and dissolved calcium as important niche axes, we estimated niche boundaries using the SVDD to identify a large number of lakes in the Great Lakes region as vulnerable to invasion by zebra mussels. The accuracy of the estimated model was very good by disciplinary standards. Particularly, the observed area-under-curve statistic (0.88) was higher than in any comparable study of which we are aware (e.g., Elith et al. 2006); cf. discussion [Drake et al. 2006]).

Other studies have previously taken a landscape approach to quantifying zebra mussel invasion risk (Strayer 1991; Drake and Bossenbroek 2004; Whittier et al. 2008). The picture emerging from these analyses is that landscape scale management may halt or slow the westward spread of this nuisance species. Accordingly, multi-state projects like the 100th Meridian Initiative (http://100thmeridian.org/) have been developed to intercept zebra mussels and respond rapidly when infested lakes are detected. However, within the continent-wide potential distribution of zebra mussels there clearly will be unsuitable ecosystems, pockets of non-habitat, and other lakes that are highly vulnerable. Bioeconomic models have shown that prevention and control of zebra mussels at the ecosystem scale would be expedient (Leung et al. 2002; Bossenbroek et al. 2009), but until now a simple, reliable model for profiling lake vulnerability based on data from North America has not been available. Further, our approach is not limited to zebra mussels but could be used for forecasting invasion of any species in which habitat availability is at least a partial determinant of spread.

Our approach accommodates the presence-only and spatial autocorrelation problems that have often afflicted niche estimation. Our approach does not solve the problem, presented by coupled source-sink habitat patches, in which organisms are found in environments that do not belong to the niche (Gomulkiewicz et al. 1999; Pulliam 2000). In our case, the data are unlikely to be contaminated by such false positive observations because zebra mussels are rarely observed before a self-sustaining local population is established. Though this fortunate situation will not generally occur, we remark that since source habitats and sink habitats must be spatially coupled, a signature of the niche boundary will almost certainly remain in the correlations among environments in which species are observed. This will not be a simple relationship, as the source-sink dynamic further erodes the structure of an already problematic dataset. We regard this as an open problem for statistical learning.

To conclude, we remark that the SVDD approach to niche estimation could be applied to other conservation problems and in basic research. As an element of population viability analysis for threatened and endangered species, niche modeling could be used for habitat assessment (Elith and Burgman 2003). Similarly, risk assessment for release or escape of genetically modified organisms should identify the eventual range such species might occupy (Wolfenbarger and Phifer 2000). But, at best, only experimental data (not geographic distribution) will be available to such models. The non-probabilistic nature of the support vector machine admits such contrived data. Further, combined with models for regional and global climate change, our approach to niche modeling could be used to predict future species distributions, particularly since statistical downscaling of course-grained models could be embedded within the statistical learning process. Finally, combined with resampling approaches to hypothesis testing (e.g., Manly 1991), these methods could be used to shed light on such longstanding questions as the relative importance of environment versus species interactions and community drift in determining species distributions and abundance (Hubbell 2001), adaptation (Gomulkiewicz et al. 1999), evolution of species ranges (Kirkpatrick and Barton 1997), niche conservatism (Holt 1996), and habitat effects on population dynamics in heterogeneous landscapes (Engen et al. 2002). The problems of presence-only and autocorrelation data have made these questions difficult to resolve.