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Jameson Cook
Jameson Cook

Classification And Regression Trees Breiman Pdfl



Bagging predictors is a method for generating multiple versions of a predictor and using these to get an aggregated predictor. The aggregation averages over the versions when predicting a numerical outcome and does a plurality vote when predicting a class. The multiple versions are formed by making bootstrap replicates of the learning set and using these as new learning sets. Tests on real and simulated data sets using classification and regression trees and subset selection in linear regression show that bagging can give substantial gains in accuracy. The vital element is the instability of the prediction method. If perturbing the learning set can cause significant changes in the predictor constructed, then bagging can improve accuracy.




Classification And Regression Trees Breiman Pdfl



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.


Adding one further step of randomization yields extremely randomized trees, or ExtraTrees. While similar to ordinary random forests in that they are an ensemble of individual trees, there are two main differences: first, each tree is trained using the whole learning sample (rather than a bootstrap sample), and second, the top-down splitting in the tree learner is randomized. Instead of computing the locally optimal cut-point for each feature under consideration (based on, e.g., information gain or the Gini impurity), a random cut-point is selected. This value is selected from a uniform distribution within the feature's empirical range (in the tree's training set). Then, of all the randomly generated splits, the split that yields the highest score is chosen to split the node. Similar to ordinary random forests, the number of randomly selected features to be considered at each node can be specified. Default values for this parameter are p \displaystyle \sqrt p for classification and p \displaystyle p for regression, where p \displaystyle p is the number of features in the model.[17]


Random forests can be used to rank the importance of variables in a regression or classification problem in a natural way. The following technique was described in Breiman's original paper[9] and is implemented in the R package randomForest.[10]


Instead of decision trees, linear models have been proposed and evaluated as base estimators in random forests, in particular multinomial logistic regression and naive Bayes classifiers.[5][27][28] In cases that the relationship between the predictors and the target variable is linear, the base learners may have an equally high accuracy as the ensemble learner.[29][5]


Mentch and Zhou (2020a) were critical of this explanation, noting in particular that in regression settings, the opposite appears to be true: random forests appear to be most useful in settings where the signal-to-noise ratio (SNR) is low. The authors proposed an alternative explanation based on degrees of freedom wherein the additional node-level randomness in random forests implicitly regularizes the procedure, making it particularly attractive in low SNR settings. Based on this idea, they demonstrate empirically that the advantage of random forests over bagging is most pronounced at low SNRs and disappears at high SNRs; when the trees in random forests are replaced with linear models, these kind of implicit [End Page 116] regularization claims and can be proved formally. This finding deals a serious blow to the widely held belief shared by both Breiman (2001) and Wyner et al. (2017) that random forests simply "are better" than bagging as a general rule. Perhaps most surprisingly, the authors also demonstrate that when a linear model selection procedure like forward selection incorporates a similar kind of random feature availability at each step, the resulting models can be substantially more accurate and even routinely outperform classical regularization methods like the lasso.


Decision trees, or classification trees and regression trees, predict responses to data. To predict a response, follow the decisions in the tree from the root (beginning) node down to a leaf node. The leaf node contains the response. Classification trees give responses that are nominal, such as 'true' or 'false'. Regression trees give numeric responses.


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.


We address the problem of selecting and assessing classification and regression models using cross-validation. Current state-of-the-art methods can yield models with high variance, rendering them unsuitable for a number of practical applications including QSAR. In this paper we describe and evaluate best practices which improve reliability and increase confidence in selected models. A key operational component of the proposed methods is cloud computing which enables routine use of previously infeasible approaches.


We describe in detail an algorithm for repeated grid-search V-fold cross-validation for parameter tuning in classification and regression, and we define a repeated nested cross-validation algorithm for model assessment. As regards variable selection and parameter tuning we define two algorithms (repeated grid-search cross-validation and double cross-validation), and provide arguments for using the repeated grid-search in the general case.


We show results of our algorithms on seven QSAR datasets. The variation of the prediction performance, which is the result of choosing different splits of the dataset in V-fold cross-validation, needs to be taken into account when selecting and assessing classification and regression models.


Hastie et al.[4] devote a whole chapter in their book to various methods of selecting and assessing statistical models. In this paper we are particularly interested in examining the use of cross-validation to select and assess classification and regression models. Our aim is to extend their findings and explain them in more detail.


Methodological advances in the last decade or so have shown that certain common methods of selecting and assessing classification and regression models are flawed. We are aware of the following cross-validation pitfalls when selecting and assessing classification and regression models:


We demonstrate the effects of the above pitfalls either by providing references or our own results. We then formulate cross-validation algorithms for model selection and model assessment in classification and regression settings which avoid the pitfalls, and then show results of applying these methods on QSAR datasets.


The contributions of this paper are as follows. First, we demonstrate the variability of cross-validation results and point out the need for repeated cross-validation. Second, we define repeated cross-validation algorithms for selecting and assessing classification and regression models which deliver robust models and report the associated performance assessments. Finally, we propose that advances in cloud computing enable the routine use of these methods in statistical learning.


In stratified V-fold cross-validation the output variable is first stratified and the dataset is pseudo randomly split into V folds making sure that each fold contains approximately the same proportion of different strata. Breiman and Spector [5] report no improvement from executing stratified cross-validation in regression settings. Kohavi [6] studied model selection and assessment for classification problems, and he indicates that stratification is generally a good strategy when creating cross-validation folds. Furthermore, we need to be careful here, because stratification de facto breaks the cross-validation heuristics.


We applied cross-validation for parameter tuning in classification and regression problems. How do we choose optimal parameters? In some cases the parameter of interest is a positive integer, such as k in k-nearest neighbourhood or the number of components in partial-least squares, and possible solutions are 1,2,3,.. etc. In other cases we need to find a real number within some interval, such as the cost value C in linear Support Vector Machine (SVM) or the penalty value λ in ridge regression. Chang and Lin [7] suggest choosing an initial set of possible input parameters and performing grid search cross-validation to find optimal (with respect to the given grid and the given search criterion) parameters for SVM, whereby cross-validation is used to select optimal tuning parameters from a one-dimensional or multi-dimensional grid. The grid-search cross-validation produces cross-validation estimates of performance statistics (for example, error rate) for each point in the grid. Dudoit and van der Laan [8] give the asymptotic proof of selecting the tuning parameter with minimal cross-validation error in V-fold cross-validation and, therefore, provide a theoretical basis for this approach. However, the reality is that we work in a non-asymptotic environment and, furthermore, different splits of data between the folds may produce different optimal tuning parameters. Consequently, we used repeated grid-search cross-validation where we repeated cross-validation Nexp times and for each grid point generated Nexp cross-validation errors. The tuning parameter with minimal mean cross-validation error was then chosen, and we refer to it as the optimal cross-validatory choice for tuning parameter. Algorithm 1 is the repeated grid-search cross-validation algorithm for parameter tuning in classification and regression used in this paper:


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