Breiman Classification And Regression Trees Ebook Download
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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. This allows to emphasise the peculiar methodological aspects that a general segmentation procedure should achieve.
Identifying and characterizing how mixtures of exposures are associated with health endpoints is challenging. We demonstrate how classification and regression trees can be used to generate hypotheses regarding joint effects from exposure mixtures.
In this paper we describe how classification and regression trees (C&RT) can be used as an alternative method for identifying complex joint effects, including interactions, for multiple exposures. The proposed approach expands the applicability of C&RT to epidemiologic research by demonstrating how it can be used for risk estimation. We view this method as a means to generate hypotheses about joint effects that may merit further investigation. We illustrate this approach with an investigation of the effect of outdoor air pollutant concentrations on emergency department visits for pediatric asthma.
Perhaps the most important way in which the proposed algorithm differs from available C&RT programs is in its control for confounding. Rarely in observational epidemiologic research are we immune to the hazards of confounding. Nonetheless, because most C&RT programs were developed for the purposes of prediction and classification, and not causal inference, they do not directly account for confounding. The typical C&RT approach is to consider all covariates one-at-a-time in the search for the optimal split [7]; however, this one-at-a-time approach ignores confounding. One approach for handling confounding is to first remove the association with the confounders and then fit a regression tree to the residuals [15]; unfortunately, this approach is appropriate only for Gaussian outcomes and cannot be easily applied to the residuals from generalized linear models (e.g. binomial or Poisson data) [16]. Conditional inference trees, first proposed by Hothorn et al. in 2006, offer a framework for recursive partitioning in which the best split is chosen conditional on all possible model splits [17]; however, this approach requires that all covariates in the conditional model be eligible for partitioning. The C&RT algorithm we propose differentiates exposure covariates from control covariates, i.e., it allows for user-defined a priori control of confounding while restricting the selection of the optimal splits to the exposure covariates, thereby making this approach better aligned to epidemiologic research when effect estimation is of interest. Bertolet et al. identified many of the same limitations to the existing C&RT approaches and go on to present a similar method for using classification and regression trees that control for confounding with Cox proportional hazards models and survival data [18].
While most C&RT packages utilize measures of node impurity, including the Gini index for classification trees and least squares for regression trees [7], to guide the splitting decisions there are situations in which other criteria may be justifiable. One approach is to base the best split on statistical significance, as was done in this paper and has been favored by others [17, 18]. Selecting splits based on the smallest P- value (or largest Chi-square statistic) illustrates how recursive partitioning can be used to capture the strongest association present in the data.
With advances in science and technology, high dimensional datasets are increasingly common, leading many researchers to question how best to characterize and analyze these mixtures of exposures. Many issues arise when dealing with mixtures, including exposure covariation, physiological and chemical interaction, joint effects, and novel exposure metrics. Classification and regression trees offer an alternative to traditional regression approaches and may be well-suited for identifying complex patterns of joint effects in the data. While recursive partitioning approaches such as C&RT are not new, they are seldom used in epidemiologic research. We believe that the aforementioned modifications to the C&RT algorithm, namely the differentiation of exposure and control covariates to account for confounding and the withholding of a referent group, can aid researchers interested in generating hypotheses about exposure mixtures.
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:
We sought to analyse and improve upon the existing cross-validation practices in selection and assessment of regression and classification models. No single cross-validation run provided for reliable selection of the best model on those datasets. Robust model selection required summarising the loss function across multiple repetitions of cross-validation. The model selection behaviour of a particular dataset could only be discerned upon performing the repeated cross-validation. Our model selection was based on average loss. 153554b96e
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