| Ecologists readily accept the concept of complex species response functions to
multiple interacting factors. In representing those species responses, however, ecologists
usually fall back on simplistic statistical models that cannot hope to capture the
complexity of a species in relationship to its habitat. The models usually lack
interaction terms and the default response shapes are typically linear (as in multiple
linear regression) or sigmoid (as in logistic regression). Yet the viewpoint most widely
accepted among ecologists is that species have hump-shaped response functions to
environmental gradients. Furthermore, we expect the shape of this response to depend on
other factors. In other words, factor interactions should be expected. Linear models may
be appropriate in some cases, such as species responses to short environmental gradients.
Likewise, logistic response functions are sometimes appropriate, for example, with a
sigmoid relationship between a species probability of occurrence and a successional
gradient. Many other possibilities exist, however (see illustration below). The standard
ecological concept for the relationship of species to environmental gradients is a
unimodal, hump-shaped curve, such as those popularized by Whittaker. Though widely
accepted as a theoretical model, where are the statistical models of single-species
response functions that incorporate a hump-shaped response? These are surprisingly rare in
the ecological literature. Even more rare are models where the shape and size of the
unimodal response depends on another variable, yet this should be the norm.
The challenge for habitat modeling is exactly the same as that expressed for data
analysis in general by Scott (1992, p. 5): "The modern challenge in data analysis is
to be able to cope with whatever complexities may be intrinsic to the data. The data may,
for example, be strongly non-Normal, fall onto a nonlinear subspace, exhibit multiple
modes, or be asymmetric [all of these are commonly true of species responses]. Dealing
with these features becomes exponentially more difficult as the dimensionality of the data
increases, a phenomenon known as the curse of dimensionality."
This curse applies to all data sets on species performance in relation to multiple
habitat factors. As the number of factors increases, the number of potential interaction
terms increases exponentially. The number of transformations or combinations of
transformations similarly inflates. The number of possible combinations of factors to
include or exclude balloons.
Huston (2002) described well the some of the pointless arguments and faulty conclusions
that have emerged from using simple, inappropriate statistical models to represent complex
systems of interacting factors. He encouraged us to recognize that "the interactive
effects of multiple limiting factors require new statistical approaches for quantifying
ecological processes..."
HyperNiche offers a new approach to solving this problem: multiplicative models.
Nonparametric Multiplicative Regression (NPMR; McCune 2006) effectively represents species
responses to multiple habitat factors. Factor interactions are accommodated automatically
and the overall form of the response surface need not be specified in advance. With a
built-in cross-validation procedure to reduce problems of overfitting, NPMR promises
models that fit better and are more parsimonious than traditional methods.
With multiplicative models, the effect of each variable can depend on the value of
other variables. This is simple conceptually but mathematically difficult for traditional
modeling methods. A practical solution is provided by adapting nonparametric curve fitting
techniques, the components being combined multiplicatively rather than additively - this
is NPMR.
The main purpose of HyperNiche is to offer this practical solution. HyperNiche
incorporates NPMR into an easy-to-use yet powerful package, complete with 3D and 2D
graphics, GIS interface, easy connections to spreadsheets and community analysis, data
transformation, and predictions for new cases.
References
Huston, M. A. 2002. Critical issues for improving predictions. Pages 7-21 in J. M.
Scott, P. J. Heglund, M. L. Morrison, J. B. Haufler, M. G. Raphael, W. A. Wall, & F.
B. Samson, eds., Predicting Species Occurrences: Issues of Accuracy and Scale. Island
Press, Washington.
McCune,
B. 2006. Non-parametric habitat models with automatic interactions. Journal of Vegetation
Science 17: 819-830
Scott, D. W. 1992. Multivariate Density Estimation: Theory, Practice, and Visualization.
John Wiley, New York. 317 pp. |