MECHANISED TUNNELLING | TECHNICAL
(response) and independent variables (predictors) is not exactly known before building a predictive model (Breiman et al. 1984). Furthermore, there is no need to consider prior suppositions about the relationship between variables. Given a set of samples, CART identifies one input
variable and one break-point, before partitioning the samples into two child nodes. Starting from the entire set of available training samples (root node), recursive binary partition is performed for each node until no further split is possible or a certain terminating criterion is satisfied. At each node, best split is identified by exhaustive
search, i.e., all potential splits on each input variable and each break-point are tested, and the one corresponding to the minimum deviations by, respectively, predicting two child nodes of samples with their mean output variables is selected. After the tree growing procedure, typically an overly
large tree is constructed, resulting in lack of model generalisation to unseen samples. A procedure of pruning is then employed to remove sequentially those splits insufficiently contributing to training accuracy. The tree is pruned all the way back to the root node, resulting in a sequence of candidate trees, each of which is tested on an independent validation sample set. The candidate tree corresponding to the lowest prediction error is selected as the final tree (Breiman 2001; Wu et al. 2008; Yang et al. 2017; Salimi 2021). To perform the RT analysis, recursive partitioning
and multiple regressions are carried out. From the root node, the data splitting process is repeated until a stop condition previously specified is reached. Stopping criteria can be defined for performing the regression tree algorithm – to keep the resultant tree from being too complicated for interpretation – and the three main ones are: (1) a minimum number of observations in a node split; (2) the depth of the tree; and, (3) the complexity parameter (cp). The complexity is the value analysed by the following rule: if the division in a specified node does not improve the model fit (on the value of set cp) then it is ignored (Therneau and Atkinson 1997; Therneau et al. 2012; Tomczyk and Ewertowski 2013; Salimi 2021). Alternatively, the optimal tree structure can be
identified through tenfold cross-validation. In brief, in regression problems, where the output is a continuous number, CART can successfully predict the targeted outcome through using, for example, least absolute deviation (LAD) error. The study discussed here used CART RT models implemented in the R statistical computing software. There is often a balance to be achieved in the depth
and complexity of the tree to optimise predictive performance on some unseen data. Respective to tree depth, the higher the depth the more complicated is the model and harder to produce the tree; if the depth is low, the efficiency of the model will be reduced and some parameters may be omitted. Hence, the related tree depth was reduced to 5, 6, 7 and 8.
To find this balance, one typically grows a very large
tree, which is then pruned back to find an optimal subtree using a cost complexity parameter that penalises our objective function LAD for the number of terminal nodes. For a given value of the smallest pruned tree that has the lowest penalised error, the optimum setting can be achieved. Smaller penalties tend to produce more complex models, which result in larger trees; larger penalties result in much smaller trees. Behind the scenes, CART RT via R computing software, is applied with a range of cost complexity (cp) values
Above, figure 6:
Comparison between the calculated and predicted FPI based on rock type categorisation via
regression tree (CART) for training and testing datasetscategorisation via regression analysis for training and testing datasets
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