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Dimension reduction methods are types of data pre-processing that are essential when dealing with large number of variables and/or collinearity. One form of dimension reduction is variable selection and it sometimes is also referred as feature selection. Variable selection involves selecting the most important or useful sets of variables from a multi-variable dataset in order to maximize predictive power (Guyon & Elisseeff, 2003; Dormann et al., 2013). While there are supervised and unsupervised methods of variable selection, it’s usually supervised methods that are used in species distribution modelling by utilising species presence/absence points as training data to select variables based on individual importance as well as magnitude of variable interactions. Among the many variable selection (and ranking) methods Pearson’s correlation coefficient backed by expert

18 _{Recent access: 11.02.2014 22:45 New Zealand time}

19 _{The DEM data was originally accessed from SRTM and GTOPO30 projects by Hijmans et al. (2005a)}

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knowledge, decision trees and the Information theoretic criteria are frequently used in species distribution modelling (Dormann et al., 2008; Elith et al., 2013). Variable selection preserves the original units and magnitudes of the variables (unless normalized) that is preferable when understanding the direct effect of each variable is important (Guyon & Elisseeff, 2003). However, there are cases where data space dimension reduction (feature construction) becomes necessary. Space dimension reduction/feature construction replaces a number of given input variables with fewer artificial variables that are constructed out of the original variables. There are two major reasons for such data processing, one is to be able to extract the most information from the input variables without involving prior knowledge about variable importance or collinearity among variables. The other is to increase predictive power by revealing data patterns that are difficult to characterise usually due to multi-collinearity in the original dataset. In case of the latter, supervised reduction methods that construct features that are relevant to the study are used. For example, using variables selected based on presence points of a specific species as a base for dimension reduction, reveals features that explain variance in the presence dataset better than a number of variables unrelated to the study system (Guyon & Elisseeff, 2003).

In this study, two commonly used methods and one new method will be used to investigate the effect of dimension reduction on accuracy of species distribution model results. The first method was to select subsets of the most important variables according to a two-step variable selection that involves random forests and step-wise regression. The second method was to construct principal components through linear transformation of the variables using principal component analysis (PCA). The third method was to produce non- linear transformed artificial components using the hierarchical-bottleneck neural network (also known as auto-associative neural network) algorithm (Scholz & Vigario, 2002) that performs hierarchical non-linear principal component analysis (h-NLPCA).

4.2.3.1 Two-step random forest / stepwise regression (DR1)

The random forest algorithm was selected for the first step of variable selection as it can handle a dataset with large number of variables similar to the ones used in this study. The random forest classifier results in low bias selection by averaging over a large ensemble of

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high-variance but low correlation trees (Breiman, 2001; Worner et al., 2010). Random forests have been used both as variable selection as well as a main species distribution model with highly satisfying results (Garzon et al., 2006; Buisson et al., 2010; Kampichler et al., 2010; Lorena et al., 2011; Senay et al., 2013; Worner et al., 2014). Stepwise regression was used to further select the most predictive subset of variables from those selected by the random forest classifier. It is important to mention that there is a caution against using stepwise selection methods (Anderson, 2007). However, its use here is justified because of the insignificant difference reported between a full model space search and the step-wise method in a similar factorial study with even more factors by Dormann et al. (2008) and the robustness of the random forest classifier used at the first step. The Akaike’s Information Criteria (AIC) was used to rank variable importance in this two-step variable selection method. The second step was added to ensure too many variables are not selected. The two- step approach was used because step-wise regression is shown to be unstable when used with large number of variables, hence applying the first step to prune certain variables of less importance with a random forest classifier which is shown to have low bias in variable selection.

4.2.3.2 Principal component analysis (DR2)

The second dimension reduction method used was a principal component analysis (PCA). PCA is a mathematical method that transforms a set of raw variables into linearly uncorrelated variables by mapping the newly transformed data on artificial orthogonal axes (Pearson, 1901). Each new variable from the transformed data are known as principal components of the data where the first principal component explains most of the variance in the data followed by the second principal component explaining most of the remaining variance provided that this data is orthogonal to the first component. The remaining variance is mapped in the same manner where subsequent principal components explain reduced variance (Legendre & Legendre, 2012). The PCA itself or slightly modified versions have been used in ecological modelling either as a dimension reduction method or as the main species distribution model (Hirzel et al., 2002; Dupin et al., 2011). Although results pertaining to data treated with PCA not being significantly different from data with no

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dimension reduction method were reported by Dormann et al. (2008), the method is included in this experiment because: 1) it is the most used space dimension reduction method in species distribution model studies when one is used, 2) it is important to replicate the result for different species data as well as more models as recommended by Dormann et al. (2008).

4.2.3.3 Non-linear principal component analysis (DR3)

The third dimension reduction method used was a hierarchical non-linear principal component analysis (h-NLPCA)21_{, a neural network model developed by Scholz and Vigario}

(2002). While there are many non-linear PCA methods like symmetrical NLPCA (s-NLPCA) (Kramer, 1991), Principal Curves (Hastie & Stuetzle, 1989), Kernel PCA22 _{(Schölkopf et al.,}

1998) etc. the h-NLPCA was chosen because it is reported to be the true non-linear extension of the linear PCA, as it achieves a hierarchical order of principal components similar to the linear PCA (Gorban, 2007). Generally, the h-NLPCA is both scalable and stable similar to the linear PCA method (Appendix 4.3).

Most of the h-NLPCA parameters are internally computed as they get adjusted throughout the iterative learning. Network weights were initialized at random. The weight decay was set at 0.001; maximum iteration was conditionally set by taking either five times the number of observations or 3000 whichever is the minimum.

21_{http://www.nlpca.org/ recent access: 12/02/2014 17:32 NZ time}

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Figure 4.3: The network topology of the hierarchical auto-associative neural network with bottleneck architecture used for the h-NLPCA

the left side of the red line in the middle show the first part of the dimension reduction process where data are extracted non-linearly from the inputs in [x1, x2, x3 …..] and linearly decoded at [z1, (z2)] (function Φextr); the right side of the red line shows where data is linearly decoded from [z1, (z2)] and are non-linearly generated at the output [x1, x2, x3…] (function Φgen). Illustration and description from Scholz et al. (2008, pp. 49-50)