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Regression trees are supervised learning methods that address multiple regression problems. The obtained models consist of a hierarchy of logical tests on the values of any of the p predictor variables. The terminal nodes of these trees, known as the leaves, contain the numerical predictions of the model for the target variable Y.
Regression trees are supervised learning methods that address multiple regression problems. The obtained models consist of a hierarchy of logical tests on the values of any of the p predictor variables.
The terminal nodes of these trees, known as the leaves, contain the numerical predictions of the model for the target variable Y. This book has established several standards in many theoretical aspects of tree-based regression, including over-fitting avoidance by post-pruning, the notion of surrogate splits for handling unknown variable, and estimating variable importance. Regression trees have several features that make them a very interesting approach to several multiple regression problems.
In spite of all these advantages, regression trees have poor prediction accuracy in several domains because of the piecewise constant approximation they provide. Using a regression tree for obtaining predictions for new observations is straightforward. For each new observation a path from the root node to a leaf is followed, selecting the branches according to the variable values of the observation.
The leaf contains the prediction for the observation. If the termination criterion is not met by the input sample D , the algorithm selects the best logical test on one of the predictor variables according to some criterion. This test divides the current sample into two partitions: the one with the cases satisfying the test, and the remaining. The algorithm proceeds by recursively applying the same method to these two partitions to obtain the left and right branches of the node.
The choices for these components are related to the preference criteria that are selected to build the trees. The most common criterion is the minimization of the sum of the square errors, known as the least squares LS criterion. With respect to the termination criterion , usually very relaxed settings are selected so that an overly large tree is grown. The reasoning is that the trees will be pruned afterward with the goal of overcoming the problem of over-fitting of the training data.
Finding the best split test for a node t involves evaluating all possible tests for this node using Eq. For each predictor of the problem one needs to evaluate all possible splits in that variable.
For continuous variables this requires a sorting operation on the values of this variable occurring in the node. Departures from the standard learning procedure described above include, among others: the use of multivariate split nodes e. An alternative is to stop tree growth sooner in a process known as pre-pruning, which again needs to be guided by reliable error estimation to know when over-fitting is starting to occur.
Although more efficient in computational terms, this latter alternative may lead to stop tree growth too soon even with look-ahead mechanisms. Post-pruning is usually carried out in a three stages procedure: a a set of sub-trees of the initial tree is generated; b some reliable error estimation procedure is used to obtain estimates for each member of this set; and c some method based on these estimates is used to select one of these trees as the final tree model.
Different methods exist for each of these steps. A common setup e. The final tree is selected using the x -SE rule, which starts with the lowest estimated error sub-tree and then selects the smallest tree within the interval of x standard errors of the lowest estimated error tree a frequent setting is to use one standard error.
Encyclopedia of Machine Learning Edition. Editors: Claude Sammut, Geoffrey I. Contents Search. Regression Trees. Download entry PDF. How to cite. The most common regression trees are binary with logical tests in each node an example is given on the left graph of Fig.
All observations in a partition are predicted with the same constant value, and that is the reason for regression trees sometimes being referred to as piecewise constant models. Open image in new window. Breiman, L. Classification and regression trees. Google Scholar. General estimates of the intrinsic variability of data in nonlinear regression models.
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Classification and Regression Trees, by Leo Breiman,. Jerome H. Friedman, Richard A. Olshen, and Charles. J. Stone. Brooks/Cole Publishing, Monterey,
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Statistical Science, 16 (3), pp , • L Breiman, JH Friedman, RA Olshen, and CJ Stone. Classification and Regression Trees. Wadsworth Inc,