File Name: classification and regression trees breiman .zip
Decision trees are then used to classify new data. In order to use CART we need to know number of classes a priori. For building decision trees, CART uses so-called learning sample -a set of historical data with pre-assigned classes for all observations. For example, learning sample for credit scoring system would be fundamental information about previous borrows variables matched with actual payoff results classes. Decision trees are represented by a set of questions which splits the learning sample into smaller and smaller parts.
A possible question could be: "Is age greater than 50? CART algorithm will search for all possible variables and all possible values in order to find the best split -the question that splits the data into two parts with maximum homogeneity.
The process is then repeated for each of the resulting data fragments. Here is an example of simple classification tree, used by San Diego Medical Center for classification of their patients to different levels of risk:In practice there can be much more complicated decision trees which can include dozens of levels and hundreds of variables. As it can be seen from figure 1.
Among other advantages of CART method is its robustness to outliers. Usually the splitting algorithm will isolate outliers in individual node or nodes. An important practical property of CART is that the structure of its classification or regression trees is invariant with respect to monotone transformations of independent variables. One can replace any variable with its logarithm or square root value, the structure of the tree will not change. Classification treeClassification trees are used when for each observation of learning sample we know the class in advance.
Classes in learning sample may be provided by user or calculated in accordance with some exogenous rule. For example, for stocks trading project, the class can be computed as a subject to real change of asset price.
Let t p be a parent node and t l ,t r -respectively left and tight child nodes of parent node t p. Consider the learning sample with variable matrix X with M number of variables x j and N observations. Let class vector Y consist of N observations with total amount of K classes. Classification tree is built in accordance with splitting rule -the rule that performs the splitting of learning sample into smaller parts.
We already know that each time data have to be divided into two parts with maximum homogeneity: where t p , t l , t r -parent, left and right nodes; x j -variable j; x R j -best splitting value of variable x j. Maximum homogeneity of child nodes is defined by so-called impurity function i t. The next important question is how to define the impurity function i t. In theory there are several impurity functions, but only two of them are widely used in practice: Gini splitting rule and Twoing splitting rule.
Gini splitting ruleGini splitting rule or Gini index is most broadly used rule. Applying the Gini impurity function 2. Ginni works well for noisy data. Besides mentioned Gini and Twoing splitting rules, there are several other methods.
But it has been proved 1 that final tree is insensitive to the choice of splitting rule. It is pruning procedure which is much more important. We can compare two trees, build on the same dataset but using different splitting rules. Squared residuals minimization algorithm is identical to Gini splitting rule. Gini impurity function 2. Maximum tree may turn out to be very big, especially in the case of regression trees, when each response value may result in a separate node.
Next chapter is devoted to different prunning methods -procedure of cutting off insignificant nodes. Choice of the Right Size TreeMaximum trees may turn out to be of very high complexity and consist of hundreds of levels. Therefore, they have to be optimized before being used for classification of new data.
Tree optimization implies choosing the right size of tree -cutting off insignificant nodes and even subtrees. Two pruning algorithms can be used in practice: optimization by number of points in each node and cross-validation.
Optimization by minimum number of pointsIn this case we say that splitting is stopped when number of observations in the node is less than predefined required minimum N min. Obviously the bigger N min parameter, the smaller the grown tree. On the one hand this approach works very fast, it is easy to use and it has consistent results.
But on the other hand, it requires the calibration of new parameter N min. While defining the size of the tree, there is a trade-off between the measure of tree impurity and complexity of the tree, which is defined by total number of terminal nodes in the treeT. For the maximum tree, the impurity measure will be minimum and equal to 0, but number of terminal nodesT will be maximum.
To find the optimal tree size, one can use cross-validation procedure. Cross-validationThe procedure of cross validation is based on optimal proportion between the complexity of the tree and misclassification error. With the increase in size of the tree, misclassification error is decreasing and in case of maximum tree, misclassification error is equal to 0.
But on the other hand, complex decision trees poorly perform on independent data. Performance of decision tree on independent data is called true predictive power of the tree. Therefore, the primary task -is to find the optimal proportion between the tree complexity and misclassification error. The process repeated several times for randomly selected learning and testing samples. Although cross-validation does not require adjustment of any parameters, this process is time consuming since the sequence of trees is constructed.
Because the testing and learning sample are chosen randomly, the final tree may differ from time to time. Classification of New DataAs the classification or regression tree is constructed, it can be used for classification of new data. The output of this stage is an assigned class or response value to each of the new observations. By set of questions in the tree, each of the new observations will get to one of the terminal nodes of the tree. Dominating class -is the class, that has the largest amount of observations in the current node.
For example, the node with 5 observations of class 1, two observation of class 2 and 0 observation of class 3, will have class 1 as a dominating class.
Before applying CART to real sector, it is important to compare CART with other statistical classification methods, identify its advantages and possible pitfalls. CART algorithm will itslef identify the most significant variables and eleminate non-significant ones. To test this property, one can inlude insignificant random variable and compare the new tree with tree, built on initial dataset.
Both trees should be grown using the same parameters splitting rule and N min parameter. We can see that the final tree 5. Changing one or several variables to its logarithm or square root will not change the structure of the tree. Only the splitting values but not variables in the questions will be different. It can be seen that the structure of the tree did not change, but changed the splitting values in questions with the first variable.
Outliers can negatively affect the results of some statistical models, like Principal Component Analysis PCA and linear regression. This property is very important, because financial data very often have outliers due to financial crisises or defaults.
There are plenty of models that can not be applied to real life due to its complexity or strict assumptions. In the table 5. One can see that a dataset with 50 variables and observations is processed in less than 3 minutes. The main idea is that learning sample is consistently replanished with new observations. It means that CART tree has an important ability to adjust to current situatation in the market.
Many banks are using the Basel II credit scoring system to classify different companies to risk levels, which uses a group of coefficients and inidicators. This approach, on the other hand, requires continuous correction of all indicators and coefficients in order to adjsut to market changes. Disadvantages of CARTAs any model, method of classification and regression trees has its own weaknesses. Insignificant modification of learning sample, such as elminating several observations, could lead to radical changes in decision tree: increase or descease of tree complexity, changes in splitting variables and values.
At figure 5. In the new classification tree only x 1 participate1 in splitting questions, therefore x 2 is not considered significant anymore. Obviously, classification results of data will change with the use of new classification tree.
Therefore, instability of trees can negatively influence the financial results. In other words, all splits are perpendicular to axis. Let us consider two different examples of data structure. At the first picture 5. CART will easily handle the splits and it can be seen on the right picture.
Although, if data have more complex structure, as for example at figure 5. From example 5.
Updated: Dec 8, For classic regression trees, the model in each cell is just a constant estimate of. That is, suppose Classification and regression trees pdf. Stone, R.
Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. DOI: Breiman and J. Friedman and R. Olshen and C. Breiman , J.
Classification Algorithms and. Regression Trees. The next four paragraphs are from the book by Breiman et. al. At the university of California, San Diego Medical.
Advances in Data Science and Classification pp Cite as. The recent interest for tree based methodologies by more and more researchers and users have stimulated an interesting debate about the implemented software and the desired software. A characterisation of the available software suitable for so called classification and regression trees methodology will be described. Furthermore, the general properties that an ideal programme in this domain should have, will be defined.
Decision trees for regression ; Piecewise constant models ; Tree-based regression. Regression trees are supervised learning methods that address multiple regression problems. Skip to main content Skip to table of contents.
Skip to Main Content. A not-for-profit organization, IEEE is the world's largest technical professional organization dedicated to advancing technology for the benefit of humanity. Use of this web site signifies your agreement to the terms and conditions. Optimal partitioning for classification and regression trees Abstract: An iterative algorithm that finds a locally optimal partition for an arbitrary loss function, in time linear in N for each iteration is presented.
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