Ecological-Niche Factor Analysis: How to Compute Habitat-Suitability Maps Without Absence Data?

Ecology, 83(7), 2002, pp. 2027±2036 ᭧ 2002 by the Ecological Society of America

ECOLOGICAL-NICHE FACTOR ANALYSIS: HOW TO COMPUTE HABITAT-SUITABILITY MAPS WITHOUT ABSENCE DATA?

A. H. HIRZEL,1 J. HAUSSER,1 D. CHESSEL,2 AND N. PERRIN1,3 1Laboratory for Conservation Biology, Institute of Ecology, University of Lausanne, CH-1015 Lausanne, Switzerland 2UMR CNRS 5023, Laboratoire de BiomeÂtrie et Biologie Evolutive, UniversiteÂ Lyon I, 69622 Villeurbanne Cedex, France

Abstract. We propose a multivariate approach to the study of geographic species distribution which does not require absence data. Building on Hutchinson's concept of the ecological niche, this factor analysis compares, in the multidimensional space of ecological variables, the distribution of the localities where the focal species was observed to a reference set describing the whole study area. The ®rst factor extracted maximizes the marginality of the focal species, de®ned as the ecological distance between the species optimum and the mean habitat within the reference area. The other factors maximize the specialization of this focal species, de®ned as the ratio of the ecological variance in mean habitat to that observed for the focal species. Eigenvectors and eigenvalues are readily interpreted and can be used to build habitat-suitability maps. This approach is recommended in situations where absence data are not available (many data banks), unreliable (most cryptic or rare species), or meaningless (invaders). We provide an illustration and validation of the method for the alpine ibex, a species reintroduced in Switzerland which presumably has not yet recolonized its entire range. Key words: Capra ibex; ecological niche; GIS; habitat suitability; marginality; multivariate analysis; presence±absence data; specialization; species distribution; Switzerland.

INTRODUCTION 1) The study area is modeled as a raster map com- posed of N adjacent isometric cells. Conservation ecology nowadays crucially relies on 2) The dependent variable is in the form of presence/ multivariate, spatially explicit models in all research absence data of the focal species in a set of sampled areas requiring some level of ecological realism. This locations. includes population viability analyses (AkcËakaya et al. 3) Independent ecogeographical variables (EGV) de- 1995, AkcËakaya and Atwood 1997, Roloff and Hau¯er scribe quantitatively some characteristics for each cell. 1997), biodiversity-loss risk assessment (AkcËakaya and These may express topographical features (e.g., alti- Raphael 1998), landscape management for endangered tude, slope), ecological data (e.g., frequency of forests, species (Livingston et al. 1990, Sanchez-Zapata and nitrate concentration), or human superstructures (e.g., Calvo 1999), ecosystem restoration (Mladenoff et al. distance to the nearest town, road density). 1995, 1997), and alien-invaders expansions (Higgins 4) A function of the EGV is then calibrated so as to et al. 1999). Such studies often conjugate the power of classify the cells as correctly as possible as suitable or Geographical Information Systems (GIS) with multi- unsuitable for the species. The details of the function variate statistical tools to formalize the link between and of its calibration depend on the analysis. the species and their habitat, in particular to quantify Sampling the presence/absence data is a crucial part the parameters of habitat-suitability models. of the process. The sample must be unbiased to be Most frequently used among multivariate analyses representative of the whole population. Absence data are logistic regressions (Jongman et al. 1987, Peeters and Gardeniers 1998, Higgins et al. 1999, Manel et al. in particular are often dif®cult to obtain accurately. A 1999, Palma et al. 1999), Gaussian logistic regressions given location may be classi®ed in the ``absence'' set (ter Braak and Looman 1987, Legendre and Legendre because (1) the species could not be detected even 1998), discriminant analyses (Legendre and Legendre though it was present (McArdle 1990, Solow 1993; for 1998, Livingston et al. 1990, Manel et al. 1999), Ma- example, KeÂry [2000] found that 34 unsuccessful visits halanobis distances (Clark et al. 1993), and arti®cial were needed before one can assume with 95% con®- neural networks (Manel et al. 1999, OÈ zesmi and OÈ zes- dence that the snake Coronella austriaca was absent mi 1999, Spitz and Lek 1999). All these methods share from a given site), (2) for historical reasons the species largely similar principles: is absent even though the habitat is suitable, or (3) the habitat is truly unsuitable for the species. Only the last cause is relevant for predictions, but ``false absences'' Manuscript received 27 March 2000; revised 4 March 2001; accepted 26 July 2001; ®nal version received 12 November 2001. may considerably bias analyses. 3 Corresponding author. Here we propose a new approach speci®cally de- E-mail: [email protected] signed to circumvent this dif®culty. Requiring only 2027 2028 A. H. HIRZEL ET AL. Ecology, Vol. 83, No. 7

FIG. 1. The distribution of the focal species on any ecogeographical variable (black bars) may differ from that of the

whole set of cells (gray bars) with respect to its mean (mS ± mG), thus allowing marginality to be de®ned. It may also differ with respect to standard deviations (␴S ± ␴G), thus allowing specialization to be de®ned.

presence data as input, the Ecological-Niche Factor randomly chosen from a distribution is a priori ex- Analysis (ENFA) computes suitability functions by pected to lie that much further from the mean as the comparing the species distribution in the EGV space variance in distribution is large. The coef®cient weight-

with that of the whole set of cells. In the present paper, ing ␴G (1.96) ensures that marginality will be most we expose the concepts behind the ENFA, develop the often be between zero and one. Namely, if the global mathematical procedures required (and implemented in distribution is normal, the marginality of a randomly the software Biomapper), and illustrate this approach chosen cell has only a 5% chance of exceeding unity. through an habitat-suitability analysis of the Alpine A large value (close to one) means that the species ibex (Capra ibex). lives in a very particular habitat relative to the reference set. Note that equation (1) is given here mainly to ex- MARGINALITY,SPECIALIZATION, AND THE plain the principle of the method; the operational def- ECOLOGICAL NICHE inition of marginality implemented in our software is Species are expected to be nonrandomly distributed provided by equation (10), which is a multivariate ex- regarding ecogeographical variables. A species with an tension of (1). optimum temperature, for instance, is expected to occur Similarly, we de®ne the specialization (S) as the ratio

preferentially in cells lying within its optimal range. of the standard deviation of the global distribution (␴G)

This may be quanti®ed by comparing the temperature to that of the focal species (␴S), distribution of the cells in which the species was ob- ␴ served with that of the whole set of cells. These dis- S ϭ G . (2) tributions may differ with respect to their mean and ␴S their variances (Fig. 1). The focal species may show A randomly chosen set of cells is expected to have some marginality (expressed by the fact that the species a specialization of one, and any value exceeding unity mean differs from the global mean) and some special- indicates some form of specialization. We reemphasize ization (expressed by the fact that the species variance that speci®c values for these indexes are bound to de- is lower than the global variance). pend on the global set chosen as reference, so that a Formally, we de®ne the marginality (M) as the ab- species might appear extremely marginal or specialized solute difference between global mean (mG) and species on the scale of a whole country, but much less so on mean (mS), divided by 1.96 standard deviations (␴G)of a subset of it. the global distribution Extending these statistics to a larger set of variables ͦm Ϫ m ͦ directly leads to Hutchinson's (1957) concept of the M ϭ GS. (1) ecological niche, de®ned as a hyper-volume in the mul- 1.96␴ G tidimensional space of ecological variables within

Division by ␴G is needed to remove any bias intro- which a species can maintain a viable population duced by the variance of the global distribution: a cell (Hutchinson 1957, Begon et al. 1996). The concept is July 2002 ECOLOGICAL-NICHE FACTOR ANALYSIS 2029 used here exactly in the same sense: by ecological niche 1. Fig. 2 plots this step for a three-dimensional initial we refer to the subset of cells in the ecogeographical set. space where the focal species has a reasonable prob- To obtain the specialization factors, the reference ability to occur. This multivariate niche can be quan- system is changed in order to transform the species ti®ed on any of its axes by an index of marginality and ellipsoid into a sphere, the variance of which equals specialization. unity in each direction. In this new metrics, the ®rst Some of these axes are obviously more interesting specialization factor is the one that maximizes the var- than others, and this is why a factor analysis is intro- iance of the global distribution (while orthogonal to duced. The reasons are actually double. First, ecolog- the marginality factor). The other specialization factors ical variables are not independent. As more and more are then extracted in turn, each step removing one di- are introduced in the description, multicollinearity and mension from the space, until all V factors are extract- redundancy arise. One aim of factor analyses is to trans- ed. All specialization factors are orthogonal in the sense form V correlated variables into the same number of that the distribution of the species subset on any factor uncorrelated factors. As these factors explain the same is uncorrelated with its distribution on the others. As amount of total variance, subsequent analyses may be factors are sorted by decreasing order of specialization, restricted to the few important factors (e.g., those ex- the ®rst few (F) will thus generally contain most of the plaining the largest part of the variance) without losing relevant information. Their small number and inde- too much information. pendence make them easier to use than the original Second, specialization is expected to depend on in- EGVs, so that all following operations will be restricted teractions among variables. For instance, the temper- to them. In particular, the suitability of any cell for the ature one species prefers might vary with humidity. focal species (be it classi®ed as 0 or 1 for observation Species may thus specialize on a combination of var- data) will be calculated according to its position in the iables, rather than on every variable independently. A F-dimensional space. factor analysis may allow extraction of the linear combinations of original variables on which the focal spe- Mathematical procedures cies shows most of its marginality and specialization. Ecogeographical variables are ®rst normalized as far In Principal Component Analyses (Cooley and Lohnes as possible, e.g., through Box-Cox transformation (So- 1971, Legendre and Legendre 1998), axes are chosen kal and Rohlf 1981). Though multinormality is theo- so as to maximize the variance of the distribution. In retically needed for factor extraction through eigen- ENFA, by contrast, the ®rst axis is chosen so as to system computation (Legendre and Legendre 1998), account for all the marginality of the species, and the this method seems quite robust to deviations from nor- following axes so as to maximize specialization, i.e., mality (Glass and Hopkins 1984). EGVs are then stan- the ratio of the variance in the global distribution to dardized by retrieving means and dividing by standard that in the species distribution. deviations:

FACTOR EXTRACTION x Ϫ xÅ z ϭ ij j (3) ij ␴ Outline of the principles xj

We use raster maps, which are grids of N isometric where xij is the value of the variable xj in cell i, xÅj the mean of this variable over all cells, and ␴ its standard cells covering the whole study area. Each cell of a map xj contains the value of one variable. Ecogeographical deviation. Let Z be the N ϫ V matrix of standardized maps contain continuous values, measured for each of measurements zij. The V ϫ V covariance matrix among the V descriptive variables. Species maps contain boo- standardized variables is then computed as lean values (0 or 1), a value of 1 meaning that the 1 R ϭ ZZT (4) presence of the focal species was proved on this cell. G N A value of zero simply means absence of proof. Each cell is thus associated to a vector whose com- where ZT is the transposed matrix of Z. Because of ponents are the values of the EGV in the underlying standardization (Eq. 3), RG is also a correlation matrix. area, and can be represented by a point in the multi- The NS lines of Z corresponding to the NS cells where dimensional space of the EGVs. If distributions are the focal species was detected are then stored in a new multinormal, the scatterplot will have the shape of a NS ϫ V matrix (say S), from which the V ϫ V species hyper-ellipsoid (Fig. 2). The cells where the focal spe- covariance matrix is calculated: cies was observed constitute a subset of the global 1 distribution and are plotted as a smaller hyper-ellipsoid R ϭ SST . (5) S N Ϫ 1 within the global one. The ®rst factor, or marginality S factor, is the straight line passing through the centroids Note that in contrast to RG, RS is not a correlation of the two ellipsoids. The species marginality is the matrix, since standardization was performed on the distance between these centroids, standardized as in Eq. global data set, not on the species subset. 2030 A. H. HIRZEL ET AL. Ecology, Vol. 83, No. 7

FIG. 2. Geometrical interpretation of the Ecological-Niche Factor Analysis. Square cells of the study area are represented in a three-EGV space. The larger, lighter balloon represents the global cloud of cells, while the smaller, darker balloon represents the subset of cells where the focal species was observed. The straight line passing through their centroids (␮G and ␮S) is the marginality factor. In order to extract the variance associated with this factor, the cell coordinates are projected on a plane ␲ perpendicular to it, thereby producing the two ellipses. In reality, those operations are typically conducted with 20±30 EGVs.

Ϫ½ Ϫ½ Let u be a normed vector of the EGV space. The RRSSRG is a symmetric matrix. It can be shown that variance of the global distribution on this vector is the solution is given by the ®rst eigenvector of uTR u, while that of the species distribution is uTR u. G S TT The ®rst specialization factor should thus maximize the H ϭ (IVVϪ yy )W(I Ϫ yy ). (9) T T ratio ⌰(u) ϭ u RGu/u RSu. However, this vector must Indeed, also be orthogonal to the marginality factor m, given 1) y is an eigenvector of H because Hy ϭ (IV Ϫ as the vector of means over the V columns of S: T T yy )W (IV Ϫ yy )y ϭ 0; 1 NS 2) H is symmetrical and thus admits a base of or- T m ϭ zij . (6) thonormed eigenvectors so that Hv ϭ␭v ⇒ v y ϭ 0; N ͸ Ά·S iϭ1 and, The problem therefore becomes that of ®nding the 3) vTHv is maximum for the ®rst eigenvector, which T T T ⇒ T T vector u that maximizes ⌰(u) under the constraint m u also maximizes v Wv since v y ϭ 0 v Hv ϭ v (IV T T T ϭ 0. This is equivalent to ®nding u, such that Ϫ yy )W (IV Ϫ yy ) v ϭ v Wv.

 T The V eigenvectors of H are then back transformed, uRuS ϭ 1  Ϫ½ and the new eigenvectors (u ϭ RS v) are stored in a  umT ϭ 0 (7)  matrix U. These vectors are RS-orthogonal (all Su dis- T uRuG max. tributions have variance 1 and are uncorrelated). Fur- thermore, due to the constraint that u be orthogonal to A change in variables allows us to rewrite the problem m, this system has one null eigenvalue. The corresponding eigenvector is thus deleted from U, and m is  T vvϭ 1 substituted instead as the ®rst column. It should be vyT 0 (8) noted that, although all marginality is accounted for by  ϭ the ®rst factor, this factor is not ``pure,'' in that the  vWvT max niche of the focal species may also display some re- ½ T Ϫ½ where v ϭ RSSu, y ϭ z/͙zz , and z ϭ R m, W ϭ striction on it, in addition to its departure from the July 2002 ECOLOGICAL-NICHE FACTOR ANALYSIS 2031

mean. The amount of specialization on this ®rst axis is provided by the difference between the traces (sum of all eigenvalues) of W and H. All these procedures are implemented in the software Biomapper (A. H. Hirzel, J. Hausser, and N. Perrin, University of Lausanne, Lausanne, Switzerland). Interpretation of the factors

The coef®cients mi of the marginality factor express the marginality of the focal species on each EGV, in units of standards deviations of the global distribution. The higher the absolute value of a coef®cient, the further the species departs from the mean available habitat regarding the corresponding variable. Negative coef- FIG. 3. The suitability of any cell from the global distribution is calculated from its situation (arrow) relative to the ®cients indicate that the focal species prefers values species distribution (histogram) on all selected niche factors. that are lower than the mean with respect to the study Speci®cally, it is calculated as twice the dashed area (sum of area, while positive coef®cients indicate preference for all cells from the species distribution that lie as far or farther higher-than-mean values. An overall marginality M can from the median dashed vertical line) divided by the total number of cells from the species distribution (surface of the be computed over all EGV as histogram).

V m2 ͸ i Ίiϭ1 M ϭ (10) a factor axis. This count is normalized in such a way 1.96 that the suitability index ranges from zero to one. so that the marginalities of different species within a Practically, this is performed by dividing the species given area can be directly compared. range on each selected factor in a series of classes, in The coef®cients of the next factors receive a different such a way that the median would exactly separate two interpretation: the higher the absolute value, the more classes (Fig. 3). For every cell from the global distri- restricted is the range of the focal species on the cor- bution, we count the number of cells from the species responding variable. Note that only absolute values distribution that lay either in the same class or in any matter here, since signs are arbitrary. The eigenvalue class farther apart from the median on the same side

␭i associated to any factor expresses the amount of (Fig. 3). Normalization is achieved by dividing twice specialization it accounts for, i.e., the ratio of the var- this number by the total number of cells in the species iance of the global distribution to that of the species distribution. Thus, a cell laying in one of the two clas- distribution on this axis. Eigenvalues usually rapidly ses directly adjacent to the median would score one, decrease from the second factor to the last one, so that and a cell laying outside the species distribution would only the ®rst four or ®ve axes are useful to compute score zero. habitat suitability. Different criteria may be used for An overall suitability index of the focal cell can then the selection process, such as direct comparison with be computed from a combination of its scores on each the broken-stick distribution, or threshold value for cu- factor. In order to account for the differential ecological mulative variance. importance of the factors, we attribute equal weight to A global specialization index can be computed as marginality and specialization, but, while all the marginality component goes to the ®rst factor, the spe- V cialization component is apportioned among all factors ␭ ͸ i proportionally to their eigenvalue (the marginality fac- Ίiϭ1 S ϭ (11) tor may thus take more than half of the weight if it V also accounts for some specialization). and can be used for among-species comparisons, pro- Repeating this procedure for each cell allows to provided the same area is used as reference. duce a habitat-suitability map, where suitability values range from 0 to 1. To convert this quantitative (or semi- HABITAT-SUITABILITY MAPS quantitative) map into a presence/absence one, a A variety of methods can be envisaged to compute threshold value may be chosen, above which the cell the suitability for the focal species of any cell from the will be considered as suitable. A validation data set study area. Among the several alternatives tested, the can be used to ®nd out the best threshold value, e.g., following approach turned out to be quite robust and through the ROC plot method (Zweig and Campbell was implemented in Biomapper. It builds on a count 1993, Fielding and Bell 1997), given some a priori cost of all cells from the species distribution that lay as far values to each type of inferential error. Typically, since or farther apart from the median than the focal cell on our method builds on presence data only, lower costs 2032 A. H. HIRZEL ET AL. Ecology, Vol. 83, No. 7

TABLE 1. Nature and source of the 34 ecogeographical variables used in the Ecological-Niche Factor Analysis (ENFA) of ibex distribution.

Of®cial database Topic Source² Derived EGV AS85R cover use OFS frequency and proximity of rock, snow, forests, meadows, etc. DHM topography OFS altitude, slope, aspect, SD of altitude GWN hydrography OFS proximity of rivers and lakes Vector 200 land map OFT proximity of villages, towns, railways, roads, etc. SB ibex colonies OFEFP calibrating and validation of presence data sets ²Abbreviations: OFT, Swiss Of®ce of Topography; OFS, Swiss Of®ce of Statistics; and OFEFP, Swiss Federal Of®ce of Environment. should be attributed to cells wrongly considered as suit- quantitative variables. Frequency and distance data able than to cells wrongly considered as unsuitable. were derived from boolean variables describing soil occupancy, as of®cial data bases attribute each cell to AN APPLICATION TO THE ALPINE IBEX one category only (snow, rocks, meadow, forest, build- The alpine ibex (Capra ibex) was exterminated from ing, etc), according to a regular sampling. Distance data the Swiss Alps in the last century due to excessive express the distance between the focal cell and the hunting. Reintroduction attempts starting in 1911 were closest cell belonging to a given category. Frequency highly successful, and colonies have since grown rap- describes the proportion of cells from a given category idly throughout the Swiss Alps. As a legally protected within a circle of 1200 m radius around the focal cell. species, ibex populations were carefully monitored Circle surface is ϳ5km2, which corresponds to the since their reintroduction, so that presence data are mean area explored daily by individual ibexes (Ab- highly reliable. However, as their expansion is still hin- derhalden and Buchli 1997). dered by the patchiness of their habitat, they presum- The presence database was a digitized map of ibex ably do not occupy yet all suitable regions. Absence home ranges. The polygons drawn by fauna managers data therefore do not necessarily re¯ect poor-quality were converted in raster format at the same resolution habitat, a point which strongly advocates for the use as EGV maps. This raster was then randomly parti- of an analysis relying on presence data only. tioned into two data sets, every cell having a 0.5 chance The whole of Switzerland was chosen as reference of belonging to each set. The ®rst set (101 564 cells) area, and modeled as a raster map based on the Swiss was used to calibrate the model, and the other set Coordinate System (plane projection), comprising (101 550 cells) to validate it. Application of the ENFA 4 145 530 square cells of 1 ha (100 ϫ 100 m) each. We method to the calibration set provided an overall mar- used 34 ecogeographical variables derived from gov- ginality of M ϭ 1.1 and an overall specialization value ernmental data bases (Table 1). Topographical data (al- of S ϭ 2.2, showing that ibex's habitat differs drasti- titude, slope, and aspect) were directly obtained as cally from the mean conditions in Switzerland, and that

TABLE 2. Variance explained by the ®rst ®ve (out of 34) ecological factors, and coef®cient values for the 13 most important initial variables.

Margin- ality Spec. 1 Spec. 2 Spec. 3 Spec. 4 EGV (46%) (11%) (8%) (6%) (4%) Rock frequency 0.350 Ϫ0.105 Ϫ0.132 Ϫ0.186 Ϫ0.163 Grass frequency 0.321 Ϫ0.021 Ϫ0.044 Ϫ0.043 Ϫ0.039 Altitude 0.269 Ϫ0.365 Ϫ0.561 0.005 0.111 Distance to grass Ϫ0.245 0.021 Ϫ0.096 Ϫ0.121 Ϫ0.073 Distance to agricultural meadows 0.231 Ϫ0.062 0.017 0.588 Ϫ0.784 Frequency of Ͼ30Њ slope 0.230 Ϫ0.035 0.074 0.011 0.004 Frequency of dense forest Ϫ0.228 0.025 Ϫ0.727 0.037 Ϫ0.001 Distance to secondary roads 0.222 Ϫ0.012 Ϫ0.049 Ϫ0.129 0.115 Frequency of agricultural meadows Ϫ0.212 Ϫ0.907 Ϫ0.011 0.236 Ϫ0.383 Distance to towns 0.209 0.012 0.049 Ϫ0.021 0.002 1 SD of altitude 0.204 0.000 Ϫ0.005 Ϫ0.032 Ϫ0.044 Distance to forests 0.203 0.015 0.165 Ϫ0.068 0.108 Distance to villages 0.200 0.003 0.017 Ϫ0.097 0.320 Notes: EGVs are sorted by decreasing absolute value of coef®cients on the marginality factor. Positive values on this factor mean that ibex prefer locations with higher values on the corresponding EGV than the mean location in Switzerland. Signs of coef®cient have no meaning on the specialization factors. The amount of specialization accounted for is given in parentheses in each column heading. July 2002 ECOLOGICAL-NICHE FACTOR ANALYSIS 2033

FIG. 4. Habitat-suitability map for alpine ibex in Switzerland, as computed from ENFA. The scale on the right, marked ``HS values,'' shows the habitat suitability values represented by each shade in the map. The inset is displayed at a larger scale at bottom left, with ibex presence data on the right. In the habitat-suitability map, light shading denotes areas more suitable for ibex, and dark shading denotes areas less suitable. In the ibex presence map at bottom right, dark shading denotes ibex presence. The largest suitable patch is indeed occupied, while the smaller one is not, being either too small or unreachable.

ibex are quite restrictive on the range of conditions ϭ 0.27, frequency of slopes Ͼ30Њϭ0.23, grass fre- they withstand. The ®ve factors retained (out of the 34 quency ϭ 0.32, distance to grass ϭϪ0.24). By con- computed) accounted for 74% of the total sum of ei- trast, ibex tends to avoid forest (frequency ϭϪ0.23) genvalues (that is, 100% of the marginality and 74% and human activities (distance to secondary roads ϭ of the specialization). The marginality factor alone ac- 0.22, distance to agricultural meadows ϭ 0.23). Aspect counted for 46% of this total specialization, a quite (northness, eastness) as well as snow and water (lakes, important value, meaning that ibex display a very re- rivers) had only marginal effects. The very large ei- stricted range on those conditions for which they most- genvalue (76.6) attributed to this ®rst factor means that ly differ from background Switzerland conditions. randomly chosen cells in Switzerland are ϳ80 times Marginality coef®cients (Table 2) showed that ibexes more dispersed on this axis than the cells were ibex are essentially linked to high-altitude, steep, and rocky was recorded. Or in other words, ibex are extremely slopes, rich in pastures (rock frequency ϭ 0.35, altitude sensitive to shifts from their optimal conditions on this 2034 A. H. HIRZEL ET AL. Ecology, Vol. 83, No. 7

which is central to the whole ®eld of ecology. A basic tenet of the niche theory is that ®tness (or habitat suitability) does not bear monotonic relationships with conditions or resources, but instead decreases from either side of an optimum. In this respect, our approach differs fundamentally from other techniques like discriminant functions or ®rst-order regressions, where relationships are assumed linear and monotonic. Ac- cordingly, our analysis directly provides two key measurements regarding the niche of the focal species, namely those of marginality and of specialization. Out- puts thus have intuitive ecological meaning, and allow direct comparisons with the niche of different species. Application of the analysis to evaluate, e.g., species packaging or niche-overlap measurement among mem- bers of a guild, would be straightforward extensions of the present approach (e.g., DoleÂdec et al. 2000) FIG. 5. Box plots presenting the distributions of the habitat-suitability values for the whole set of cells (left) and the Our application to ibex data, for instance, provides validation subset (right). The latter is made of the 101 550 quantitative estimates of marginality and specialization cells with ibex that were not included in the analysis. Boxes for this species which evidence its very peculiar eco- delimit the interquartile range, the middle line indicating the logical requirements. Furthermore, interpretation of the median; whiskers encompass the 80% con®dence interval. The two distributions obviously differ, as the global one is factors in terms of the EGVs turns out to be very con- mostly con®ned to low values (suitability, 5±33%), while the sistent with the experience of ®eld specialists. In par- validation set concentrates on high-suitability values (75± ticular, the EGVs that correlate with the marginality 99%). factor are precisely those most often cited as particularly relevant for ibex ecology (Rauch 1941, Nievergelt 1966, Nievergelt and Zingg 1986, Hausser 1995, Hin- axis. The next factors account for some more special- denlang and Nievergelt 1995). ization, mostly regarding agricultural meadow fre- These results obviously suffer from the same caveat quency and altitude (second factor) as well as forest as any inferential approach: a variable might turn out frequency (third factor), showing some sensitivity to to correlate with one of the main axes not because of shifts away from their optimal values on these vari- its intrinsic importance, but because it correlates ables. strongly with another crucially important variable. A suitability map was built from these ®ve factors ENFA is a purely descriptive method and cannot extract for the whole of Switzerland, which is plotted on Fig. causality relations. Nonetheless, it provides (at worst) 4. Enlargement of a small part of it shows one suitable important cues about preferential conditions, and re- patch where ibex was indeed recorded (on the right), mains a powerful tool to draw potential habitat maps. as well as one smaller suitable patch (on the left) where In this respect, a limitation of our software is that it ibex was not recorded. This patch presumably was not does not yet include con®dence intervals on distribu- colonized because of its isolation, or too small to sus- tion maps. Increasingly, conservation managers are de- tain a viable colony on the long term. As false positives manding risk analyses that incorporate uncertainties in like this give no indication about the quality of our model predictions. These could clearly be obtained model, standard quality estimators like the kappa index through the bootstrapping of presence data. Though not (Monserud and Leemans 1992), which attribute the yet implemented in Biomapper, this procedure will cer- same importance to false positives and false negatives, tainly provide an important and useful extension. cannot be used to validate it. Instead, we evaluated the A second limitation, less easy to deal with, is that distribution of suitability values of cells from the val- ENFA only handles linear dependencies within the spe- idation set. As shown in Fig. 5, these cells differ dras- cies niche. Multiplicative or nonlinear interactions can- tically from the global distribution. Predicted suitabil- not be accommodated in the present state, except ity exceeds 0.5 in 83% of cells, which differs highly through transformations or nonlinear combinations of signi®cantly (P Ͻ 0.0001, bootstrap test) from the val- the original ecogeographical variables. ue of 24% expected if cells were randomly chosen from A third limitation is that some EGVs may turn out the global distribution. to be constant in S, or in linear combination with other DISCUSSION EGVs, which makes RS singular. This is likely to hap- pen with coarsely measured data or small species data Niche factors and distribution maps sets. Whenever this happens, Biomapper identi®es the The originality of the present approach lies in the constant or correlated EGVs so that the user can remove fact that it builds on the concept of ecological niche, (one of) them from the analysis. An alternative ap- July 2002 ECOLOGICAL-NICHE FACTOR ANALYSIS 2035 proach would obviously consist of improving the ®eld highly sensitive to the algorithms chosen, as well as to sample, either by increasing the presence data set or the input order. Consequences are that (1) many trials by measuring EGVs on a ®ner scale. are needed in order to sort out the ``best'' model, and Finally, a last important point to emphasize again is (2) variables that bear a causal relationship to the focal that our approach characterizes ecological niches rel- species' presence might well be lost in the process, if ative to a reference area. Marginality and specialization other EGVs present spurious correlations. This implies are thus bound to depend on the geographic limits of some subjective choices, and requires a good a priori the study area. Some species may turn out to occur at knowledge of the focal species' ecology. In contrast, the very edge of their distribution, and may thus appear our factor analysis does not reject any input EGV, but quite specialized in the reference set, however wide- only weights them. The subjective components and a spread they might be otherwise. Reciprocally, ibex priori knowledge required are thereby kept minimal, would have appeared much less marginal and special- and correlations among variables and axes are imme- ized had our sampling area been restricted to the Alps. diately visible and interpretable. The same distinction must be applied here as the one ACKNOWLEDGMENTS made between fundamental and realized niches (Hutch- This research was supported by the Swiss Federal Of®ce inson 1957). Our analysis does not investigate funda- for Environment, Forest and Landscape (OFEFP), grant mental niches, but only their speci®c realization within 0310.3600.305. We warmly thank H. J. Blankenhorn for his a given geographical context. enthusiastic endorsement of the Ibex Project, and all people who tested Biomapper, particularly P.Patthey for his thorough ENFA vs. logistic regressions check and judicious suggestions. With respect to more standard techniques, a crucial LITERATURE CITED advantage of ENFA is that it does not require absence Abderhalden, W., and C. Buchli. 1997. Steinbockprojekt Al- data. Presence data are compared instead with back- bris/SNP, Schlussbericht. ARINAS and FORNAT, Zernez ground environment. This of course implies that pres- (Graubunden), Switzerland. AkcËakaya, H. R., and J. L. Atwood. 1997. A habitat-based ence data should be unbiased samples of actual distri- metapopulation model of the California Gnatcatcher. Con- butions, which we suspect might not be the case of servation Biology 11:422±434. many available database, since sampling efforts are fre- AkcËakaya, H. R., M. A. McCarthy, and J. L. 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