# Regression Tree Learning Task

A Regression Tree Learning Task is a predictor tree learning task that is a model-based point estimation task (required to produce a regression tree).

**AKA:**Regression Trees, Decision Trees for Regression, Piecewise Constant Models, Tree-Based Regression Task.**Context:**- Task Input:
- A data sample [math]\displaystyle{ D=\{\langle x_i^1,\cdots, x_i^p,y_i\}^n_{i=1} }[/math].

**output**:- A Regression Tree model based on training samples.

**Task Requirements**- It (usually) requires a termination criterion for finding the best split node.

- It can be solved by a Regression Tree Learning System (that implements a Regression Tree Learning algorithm).
- It can range from being a Fully-Supervised Regression Tree Learning Task to being a Semi-Supervised Regression Tree Learning Task.
- It can be supported by a Regression Tree Post-Pruning Task and a Regression Tree Pre-Pruning Task.

- Task Input:
**Example(s):****Counter-Example(s):****See:**Regression Task, Classification Task, Trained Regression Tree Model, Kernel-based Classification Algorithm.

## References

### 2017a

- (Torgo, 2017) ⇒ Luı́s Torgo, (2017). "Regression Trees". In Encyclopedia of Machine Learning and Data Mining pp 1080-1083.
- QUOTE: Regression trees are supervised learning methods that address multiple regression problems. They provide a tree-based approximation
*f^*, of an unknown regression function [math]\displaystyle{ Y \in \mathcal{R} }[/math] and [math]\displaystyle{ \epsilon \approx N (0, \sigma^2) }[/math], based on a given sample of data [math]\displaystyle{ D=\{\langle x_i^1,\cdots, x_i^p,y_i\}^n_{i=1} }[/math]. The obtained models consist of a hierarchy of logical tests on the values of any of the [math]\displaystyle{ p }[/math] predictor variables. The terminal nodes of these trees, known as the leaves, contain the numerical predictions of the model for the target variable [math]\displaystyle{ Y }[/math]

- QUOTE: Regression trees are supervised learning methods that address multiple regression problems. They provide a tree-based approximation

### 2017b

- (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/Decision_tree_learning Retrieved:2017-10-15.
**Decision tree learning**uses a decision tree (as a predictive model) to go from observations about an item (represented in the branches) to conclusions about the item's target value (represented in the leaves). It is one of the predictive modelling approaches used in statistics, data mining and machine learning. Tree models where the target variable can take a discrete set of values are called**classification trees**; in these tree structures, leaves represent class labels and branches represent conjunctions of features that lead to those class labels. Decision trees where the target variable can take continuous values (typically real numbers) are called regression trees**.****In decision analysis, a decision tree can be used to visually and explicitly represent decisions and decision making. In data mining, a decision tree describes data (but the resulting classification tree can be an input for decision making). This page deals with decision trees in data mining.**

### 2017c

- (Furnkranz, 2017) ⇒ Johannes Fürnkranz, (2017). "Decision Tree". In Encyclopedia of Machine Learning and Data Mining pp 330-335.
- ABSTRACT: The induction of decision trees is one of the oldest and most popular techniques for learning discriminatory models, which has been developed independently in the statistical (Breiman et al. 1984
^{[1]}; Kass 1980^{[2]}) and machine learning (Hunt et al. 1966^{[3]}; Quinlan 1983^{[4]}, 1986^{[5]}) communities. A decision tree is a tree-structured classification model, which is easy to understand, even by non-expert users, and can be efficiently induced from data. An extensive survey of decision-tree learning can be found in Murthy (1998). - QUOTE: Decision trees are also often used as components in Ensemble Methods such as random forests (Breiman 2001
^{[6]}) or AdaBoost (Freund and Schapire 1996^{[7]}). They can also be modified for predicting numerical target variables, in which case they are known as regression trees. One can also put more complex prediction models into the leaves of a tree, resulting in Model Trees.

- ABSTRACT: The induction of decision trees is one of the oldest and most popular techniques for learning discriminatory models, which has been developed independently in the statistical (Breiman et al. 1984

### 2011

- (Loh, 2011) ⇒ Loh, W. Y. (2011). Classification and regression trees. Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery, 1(1), 14-23.
- ABSTRACT: Classification and regression trees are machine-learning methods for constructing prediction models from data. The models are obtained by recursively partitioning the data space and fitting a simple prediction model within each partition. As a result, the partitioning can be represented graphically as a decision tree. Classification trees are designed for dependent variables that take a finite number of unordered values, with prediction error measured in terms of misclassification cost. Regression trees are for dependent variables that take continuous or ordered discrete values, with prediction error typically measured by the squared difference between the observed and predicted values. This article gives an introduction to the subject by reviewing some widely available algorithms and comparing their capabilities, strengths, and weakness in two examples.

### 1999

- (Torgo, 1999) ⇒ Louis Torgo. (1999). “Inductive Learning of Tree-based Regression Models." Ph.D. Thesis, Thesis, Faculty of Sciences, University of Porto
- ABSTRACT: This thesis explores different aspects of the induction of tree-based regression models from data. The main goal of this study is to improve the predictive accuracy of regression trees, while retaining as much as possible their comprehensibility and computational efficiency. Our study is divided in three main parts.
In the first part we describe in detail two different methods of growing a regression tree: minimising the mean squared error and minimising the mean absolute deviation. Our study is particularly focussed on the computational efficiency of these tasks. We present several new algorithms that lead to significant computational speed-ups. We also describe an experimental comparison of both methods of growing a regression tree highlighting their different application goals.

Pruning is a standard procedure within tree-based models whose goal is to provide a good compromise for achieving simple and comprehensible models with good predictive accuracy. In the second part of our study we describe a series of new techniques for pruning by selection from a series of alternative pruned trees. We carry out an extensive set of experiments comparing different methods of pruning, which show that our proposed techniques are able to significantly outperform the predictive accuracy of current state of the art pruning algorithms in a large set of regression domains.

In the final part of our study we present a new type of tree-based models that we refer to as local regression trees. These hybrid models integrate tree-based regression with local modelling techniques. We describe different types of local regression trees and show that these models are able to significantly outperform standard regression trees in terms of predictive accuracy. Through a large set of experiments we prove the competitiveness of local regression trees when compared to existing regression techniques.

- ABSTRACT: This thesis explores different aspects of the induction of tree-based regression models from data. The main goal of this study is to improve the predictive accuracy of regression trees, while retaining as much as possible their comprehensibility and computational efficiency. Our study is divided in three main parts.

### 1963

- (Morgan & Sonquist, 1963) ⇒ Morgan, J. N., & Sonquist, J. A. (1963). Problems in the analysis of survey data, and a proposal. Journal of the American statistical association, 58(302), 415-434. DOI: 10.1080/01621459.1963.10500855
- ABSTRACT: Most of the problems of analyzing survey data have been reasonably well handled, except those revolving around the existence of interaction effects. Indeed, increased efficiency in handling multivariate analyses even with non-numerical variables, has been achieved largely by assuming additivity. An approach to survey data is proposed which imposes no restrictions on interaction effects, focuses on Importance in reducing predictive error, operates sequentially, and is independent of the extent of linearity in the classifications or the order in which the explanatory factors are introduced.

- ↑ Breiman L, Friedman JH, Olshen R, Stone C (1984) Classification and regression trees. Wadsworth & Brooks, Pacific Grove
- ↑ Kass GV (1980) An exploratory technique for investigating large quantities of categorical data. Appl Stat 29:119–127
- ↑ Hunt EB, Marin J, Stone PJ (1966) Experiments in induction. Academic, New York
- ↑ Quinlan JR (1983) Learning efficient classification procedures and their application to chess end games. In: Michalski RS, Carbonell JG, Mitchell TM (eds) Machine learning. An artificial intelligence approach, Tioga, Palo Alto, pp 463–482
- ↑ Quinlan JR (1986) Induction of decision trees. Mach Learn 1:81–106
- ↑ Breiman, L. (2001). Random forests. Machine learning, 45(1), 5-32.
- ↑ Freund Y, Schapire RE (1996) Experiments with a new boosting algorithm. In: Saitta L (ed) Proceedings of the 13th International Conference on Machine Learning, Bari. Morgan Kaufmann, pp 148–156