Linear Discriminant analysis (LDA) is an attractive means of comparing BEC variants in a reduced climate space because it emphasizes the climate variables that differentiate BEC variants. However, it is essential to note that LDA is a global classifier: the models (linear discriminating functions) maximize the ratio of total between-group to within-group variance across all groups. This “one size fits all” approach is appropriate where the processes (variables) generating the differences between groups are reasonably consistent across all groups. However, poor model fit in a reduced climate space is expected if the discriminating processes differ in various regions of the data space. This is certainly the case with the BEC climate classification.

The beauty of the BEC climate classification is that rather than defining climates in terms of predefined quantitative climate variables, it infers climate differences based on differences in vegetation communities. The climate variables creating these vegetation differences are expected to be different for each pair of adjacent BEC variants. In some cases, snow depth may be the crucial variable, in others it may be growing season frost, drought, and so on.  LDA on the entire province cannot be expected to capture the important differences between BEC variants. It is likely that any LDA solution at the provincial or even regional level will be overly reductive. There appears to be a tension between the need to represent local differences accurately and the need to understand the system as a whole. It would be desirable to have a methodology that was able to integrate a localized analysis of the differences between adjacent pairs of BEC variants with a provincial scale analysis.

Czogiel et al. (2007) provide a very nice summary of the issue:

Since the estimation is carried out without taking into account the nature of the problem at hand, i.e. the classification of a specific trial point, LDA can be considered a global classifier. Hand and Vinciotti (2003) argue that an approach like this can lead to poor results if the chosen model does not exactly reflect the underlying data generating process because then, a good fit in some parts of the data space may worsen the fit in other regions. Since in classification problems accuracy is often not equally important throughout the entire data space, they suggest to improve the fit in regions where a good fit is crucial for obtaining satisfactory results – even if the fit elsewhere is degraded. For the dichotomous logistic discrimination, two approaches have been proposed to accomplish this. Hand and Vinciotti (2003) introduce a logistic discrimination model in which data points in the vicinity of the ideal decision surface are weighted more heavily than those which are far away. Another strategy is presented by Tutz and Binder (2005) who suggest to assign locally adaptive weights to each observation of the training set. By choosing the weights as decreasing in the (Euclidian) distance to the observation to be classified and maximizing the corresponding weighted (log-)likelihood, a localized version of the logistic discrimination model can be obtained. The classifier is therefore adapted to the nature of each individual trial point which turns the global technique of logistic discrimination into an observation specific approach.

Czogiel et al. (2007) go on to describe a method they call “localized linear discriminant analysis”. This approach modifies LDA using locally adaptive weights. They state that this is a simpler approach than earlier approaches (e.g. Hand and Vinciotti 2003) based on logistic discrimination. It also is usable with multi-class situations, which logistic discrimination is not.

The introduction is compelling, but they did not provide a plain-language description of their method, and the mathematical description of their approach was over my head. The examples they provided were also not very instructive for me.

However, the basic concept appears to be relevant to the problem of fitting multi-class LDA models to local conditions. Rather than treating several BEC variants (or several hundred, as in the case of a BC-scale analysis) as equal, it might be possible to use an approach that puts a single location at the centre of the climate space classification. Instead of asking “what do BEC variants look like in climate space?”, we would be asking “what does climate space look like from the perspective of the ESSFwk1?”  discrimination (maximizing the ratio of between-group variance to within-group variance) could be weighted based on Euclidian distance of group centroids. This would favour discriminating functions that emphasize the variables responsible for differentiation of the neighbours of the focal variant in climate space.

What would this approach achieve?

1. For each BEC variant, there would be an ordered list of which climate variables differentiated it from its neighbours in climate space. This is interesting from an ecological perspective.

2. This approach would customize dimension reduction for the climatic context of an individual location. This would likely be a preferable climate space for visualizing historical and projected climate change at that location. By changing the weighting function, the realm of the climate space could be expanded or contracted as needed to incorporate large projected climate changes.

Localized LDA is an intriguing idea. It may suffer from the same flaws as discriminant analysis on an environmental gradient. On the other hand these flaws may be partially mitigated by the presence of the full set of 200+ BEC variants being available. While not as straightforward and promising as multidimensional scaling, localized LDA is worth considering as an option for analysis and visualization of climate change relative to the BEC climate classification.

References

Czogiel, I., K. Luebke, M. Zentgraf, and C. Weihs. 2007. Localized linear discriminant analysis. Pages 133–140 in R. Decker and H. Lenz, editors. 30th Annual Conference of the German-Classification-Society.

Hand, D. J., and V. Vinciotti. 2003. Local Versus Global Models for Classification Problems. The American Statistician 57:124–131.