Classification And Regression Trees Breiman Pdf
Pamula Harrison
Random forest is a commonly-used machine learning algorithm trademarked by Leo Breiman and Adele Cutler, which combines the output of multiple decision trees to reach a single result. Its ease of use and flexibility have fueled its adoption, as it handles both classification and regression problems.
Random forest algorithms have three main hyperparameters, which need to be set before training. These include node size, the number of trees, and the number of features sampled. From there, the random forest classifier can be used to solve for regression or classification problems.
Good question. @G5W is on the right track in referencing Wei-Yin Loh's paper. Loh's paper discusses the statistical antecedents of decision trees and, correctly, traces their locus back to Fisher's (1936) paper on discriminant analysis -- essentially regression classifying multiple groups as the dependent variable -- and from there, through AID, THAID, CHAID and CART models.
This response is suggesting that the arc of the evolution leading to the development of decision trees created new questions or dissatisfaction with existing "state-of-the-art" methods at each step or phase in the process, requiring new solutions and new models. In this case, dissatisfactions can be seen in the limitations of modeling two groups (logistic regression) and recognition of a need to widen that framework to more than two groups. Dissatisfactions with unrepresentative assumptions of an underlying normal distribution (discriminant analysis or AID) as well as comparison with the relative "freedom" to be found in employing nonparametric, distribution-free assumptions and models (e.g., CHAID and CART).
This 2014 article in the New Scientist is titled Why do we love to organise knowledge into trees?( -800-why-do-we-love-to-organise-knowledge-into-trees/), It's a review of data visualization guru Manuel Lima's book The Book of Trees which traces the millenia old use of trees as a visualization and mnemonic aid for knowledge. There seems little question but that the secular and empirical models and graphics inherent in methods such as AID, CHAID and CART represents the continued evolution of this originally religious tradition of classification.
This example uses a random forest (Breiman 2001) classifier with 10 trees to downscale MODIS data to Landsat resolution. The sample() method generates two random samples from the MODIS data: one for training and one for validation. The training sample is used to train the classifier. You can get resubstitution accuracy on the training data from classifier.confusionMatrix(). To get validation accuracy, classify the validation data. This adds a classification property to the validation FeatureCollection. Call errorMatrix() on the classified FeatureCollection to get a confusion matrix representing validation (expected) accuracy.
Learns a single regression tree. The procedure follows the algorithm described by "Classification and Regression Trees" (Breiman et al, 1984), whereby the current implementation applies a couple of simplifications, e.g. no pruning, not necessarily binary trees, etc.
Classification and regression trees (CART) (Breiman et al. 1984) are a popular class of machine learning algorithms. CART models seek predictors and cut points in the predictors that are used to split the sample. The cut points divide the sample into more homogeneous subsamples. The splitting process is repeated on both subsamples, so that a series of splits defines a binary tree. The target variable can be discrete (classification tree) or continuous (regression tree).
The missForest method (Stekhoven and Bühlmann 2011) successfully used regression and classification trees to predict the outcomes in mixed continuous/categorical data. MissForest is popular, presumably because it produces a single complete dataset, which at the same time is the reason why it fails as a scientific method. The missForest method does not account for the uncertainty caused by the missing data, treats the imputed data as if they were real (which they are not), and thus invents information. As a consequence, \(p\)-values calculated after application of missForest will be more significant than they actually are, confidence intervals will be shorter than they actually are, and relations between variables will be stronger than they actually are. These problems worsen as more missing values are imputed. Unfortunately, comparisons studies that evaluate only accuracy, such as Waljee et al. (2013), will fail to detect these problems.
Algorithm 3.4 describes the major steps of an algorithm for creating imputations using a classification or regression tree. There is considerable freedom at step 2, where the tree model is fitted to the training data \((\dot y_\mathrmobs,\dot X_\mathrmobs)\). It may be useful to fit the tree such that the number of cases at each node is equal to some pre-set number, say 5 or 10. The composition of the donor groups will vary over different bootstrap replications, thus incorporating sampling uncertainty about the tree.
For both classification and regression, a useful stopping criterion is to require that each split improves the relative error by at least a, a predetermined value of the complexity parameter. This parameter acts to regularize the cost function of growing the tree5 by balancing the cost with a penalty for adding additional partitions. For example, as we grow our regression tree, we monitor the relative MSE (rMSE) of each split and the amount of decrease a at each split (Fig. 2c). Splitting at X = 49 improves the rMSE by a = 0.05. However, the next candidate split at X = 22 lowers rMSE by only a = 0.007. If we use a cutoff of a = 0.01, this split would not be accepted, and tree growth would end.
Decision trees provide a framework to quantify the values of outcomes and the probabilities of achieving them. They can be used for both classification and regression problems, and create data models that will predict class labels or values for a decision-making process. The models are built from the training dataset fed to the system (supervised learning). A decision tree visualization helps outline the decisions in a way that is easy to understand, making it a popular data mining technique.
The regression technique is not very different from classification with decision trees. The main distinction is the measurement for the splitting. Instead of impurity, regression trees use mean squared error (MSE). Questions gradually reduce MSE until it reaches a minimum. The node result is the average target value from all samples in it. MSE is calculated like this:
C&RT, a recursive partitioning method, builds classification and regression trees for predicting continuous dependent variables (regression) and categorical predictor variables (classification). The classic C&RT algorithm was popularized by Breiman et al. (Breiman, Friedman, Olshen, & Stone, 1984; see also Ripley, 1996). A general introduction to tree-classifiers, specifically to the QUEST (Quick, Unbiased, Efficient Statistical Trees) algorithm, is also presented in the context of the Classification Trees Analysis facilities, and much of the following discussion presents the same information, in only a slightly different context. Another, similar type of tree building algorithm is CHAID (Chi-square Automatic Interaction Detector; see Kass, 1980).
Classification-type problems. Classification-type problems are generally those where we attempt to predict values of a categorical dependent variable (class, group membership, etc.) from one or more continuous and/or categorical predictor variables. For example, we may be interested in predicting who will or will not graduate from college, or who will or will not renew a subscription. These would be examples of simple binary classification problems, where the categorical dependent variable can only assume two distinct and mutually exclusive values. In other cases, we might be interested in predicting which one of multiple different alternative consumer products (e.g., makes of cars) a person decides to purchase, or which type of failure occurs with different types of engines. In those cases there are multiple categories or classes for the categorical dependent variable. There are a number of methods for analyzing classification-type problems and to compute predicted classifications, either from simple continuous predictors (e.g., binomial or multinomial logit regression in GLZ), from categorical predictors (e.g., Log-Linear analysis of multi-way frequency tables), or both (e.g., via ANCOVA-like designs in GLZ or GDA). The CHAID also analyzes classification-type problems, and produces results that are similar (in nature) to those computed by C&RT. Note that various neural network architectures are also applicable to solve classification-type problems.
Tree methods are nonparametric and nonlinear. The final results of using tree methods for classification or regression can be summarized in a series of (usually few) logical if-then conditions (tree nodes). Therefore, there is no implicit assumption that the underlying relationships between the predictor variables and the dependent variable are linear, follow some specific non-linear link function [e.g., see Generalized Linear/Nonlinear Models (GLZ)], or that they are even monotonic in nature. For example, some continuous outcome variable of interest could be positively related to a variable Income if the income is less than some certain amount, but negatively related if it is more than that amount (i.e., the tree could reveal multiple splits based on the same variable Income, revealing such a non-monotonic relationship between the variables). Thus, tree methods are particularly well suited for data mining tasks, where there is often little a priori knowledge nor any coherent set of theories or predictions regarding which variables are related and how. In those types of data analyses, tree methods can often reveal simple relationships between just a few variables that could have easily gone unnoticed using other analytic techniques.
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