Tree-structured Regression Research Articles

Warm Season Lightning Probability Prediction for Canada and the Northern United States

Abstract Statistical models valid May–September were developed to predict the probability of lightning in 3-h intervals using observations from the North American Lightning Detection Network and predictors derived from Global Environmental Multiscale (GEM) model output at the Canadian Meteorological Centre. Models were built with pooled data from the years 2000–01 using tree-structured regression. Error reduction by most models was about 0.4–0.7 of initial predictand variance. Many predictors were required to model lightning occurrence for this large area. Highest ranked overall were the Showalter index, mean sea level pressure, and troposphere precipitable water. Three-hour changes of 500-hPa geopotential height, 500–1000-hPa thickness, and MSL pressure were highly ranked in most areas. The 3-h average of most predictors was more important than the mean or maximum (minimum where appropriate). Several predictors outranked CAPE, indicating it must appear with other predictors for successful statistical lightning prediction models. Results presented herein demonstrate that tree-structured regression is a viable method for building statistical models to forecast lightning probability. Real-time forecasts in 3-h intervals to 45–48 h were made in 2003 and 2004. The 2003 verification suggests a hybrid forecast based on a mixture of maximum and mean forecast probabilities in a radius around a grid point and on monthly climatology will improve accuracy. The 2004 verification shows that the hybrid forecasts had positive skill with respect to a reference forecast and performed better than forecasts defined by either the mean or maximum probability at most times. This was achieved even though an increase of resolution and change of convective parameterization scheme were made to the GEM model in May 2004.

An adaptive analysis of covariance using tree-structured regression

In this article, we propose an adaptive procedure for testing for the effect of a factor of interest in the presence of one or more confounding variables in observational studies. It is especially relevant for applications where the factor of interest has a suspected causal relationship with a response. This procedure is not tied to linear modeling or normal distribution theory, and it offers a valuable alternative to traditional methods. It is suitable for applications where a factor of interest is categorical and the response is continuous. Confounding variables may be continuous or categorical. The method is comprised of two basic steps that are performed in sequence. First, confounding variables alone (i.e., without the factor of interest) are used to group observations into subsets. These subsets have the property that, when restricted to a subset, there is little or no remaining variation in the response that is attributable to the confounding variables. We then test for the factor of interest within each subset of observations. We propose to implement the first step using a technique that is generally referred to as tree-structured regression. We use a non parametric permutation procedure to carry out the second step. The proposed method is illustrated through an analysis of a U. S. Department of Agriculture (USDA) Forest Service data set and an air pollution data set.

Stabilization Of Regression Trees

In this paper, we present a hierarchical approach to simultaneous regression and classification. Regression obviously becomes more accurate by assessing a regression surface to each of a given class of a finite sample set compared to one regression surface for the sample as a whole. For class formation, a tree of regression surfaces is constructed that balances minimization of the regression error and towards unseen data. Common tree-structured regression algorithms split nodes according to an independent variable. Terminal nodes correspond to one specific class. Such constructions gave rise to less complicated approaches that have mainly been used for adaptive classification in machine learning. The considerable advantage of regression trees over a single regression is often set off by trees' poor behaviour on unseen data of supposedly the same nature. Also, a tree formation solely based on minimizing the regression error may not generate information by which to assign new data to terminal nodes. Generalization is addressed by a stabilization operation. Each sample vector is assigned to its nearest neighbour which in turn is assigned to its nearest neighbour etc. This results in so-called neighbour chains that partition the sample set. Class splitting during tree formation is here restricted to classes that contain neighbour chains completely. For classification, unseen sample data are assigned to its closest neighbour used in tree formation. The regression surface of the class containing the corresponding neighbour chain is then used for estimation. Data Mining II, C.A. Brebbia & N.F.F. Ebecken (Editors) © 2000 WIT Press, www.witpress.com, ISBN 1-85312-821-X

Interactive Tree-Structured Regression via Principal Hessian Directions

An interactive approach to tree-structured regression is introduced. Unlike other procedures driven by cost optimization, this approach focuses on the exploration of geometric information in the data. The procedure begins with finding a direction along which the regression surface bends the most. This direction is used for splitting the data into two regions. Within each region, another direction is found, and the partition is made in the same manner. The process continues recursively until the entire regressor domain is decomposed into regions wherein the surface no longer bends significantly and linear regression fit becomes appropriate. For implementing the direction search, the method of principal Hessian directions is applied. Several simulation and empirical results are reported. Comparison with three methods—CART, SUPPORT, and MARS—is made. The benefit of using geometric information is highlighted.

Interactive Tree-Structured Regression via Principal Hessian Directions

Interactive Tree-Structured Regression via Principal Hessian Directions

Application of tree-structured regression for regional precipitation prediction using general circulation model output

This study presents a tree-structured regression (TSR) method to relate daily precipita- tion with a variety of free-atmosphere variables. Historical data were used to identify distinct weather patterns associated with differing types of precipitation events. Models were developed using 67% of the data for training and the remaining data for model validation. Seasonal models were built for each of 2 US sites: San Francisco, California, and San Antonio, Texas. The average correlation between observed and simulated daily precipitation data series is 0.75 for the training set and 0.68 for the validation set. Relative humidity was found to be the dominant variable in these TSR models. Out- put from an NCAR CSM (climate system model) transient simulation of climate change were then used to drive the TSR models in the prediction of precipitation characteristics under climate change. A preliminary screening of the GCM output variables for current climate, however, revealed signifi- cant problems for the San Antonio site. Specifically, the CSM missed the annual trends in humidity for the grid cell containing this site. CSM output for the San Francisco site was found to be much more reliable. Therefore, we present future precipitation estimates only for the San Francisco site.

Extending the elements of tree-structured regression.

The usage of tree-structured or recursive partitioning methods has grown steadily in the decade since the appearance of the definitive monograph 'Classification and Regression Trees'. Accompanying this growth have been many methodologic and software extensions that have served to give the tree-structured approach even wider applicability. This overview highlights some of these developments, emphasizing the regression setting. An illustrative example describes how tree-structured regression, modified to handle right-censored, left-truncated survival data with time-dependent covariates, can be used to assess whether rates of progression from HIV to AIDS have changed over time.

Representative Curves for Longitudinal Data via Regression Trees

Abstract Simple superposition of individual curves, as arise in longitudinal studies, makes the detection of structure difficult due to clutter. This is especially true when the number of curves is large, but can even be exhibited in moderate sample settings when variability is high. Both Jones and Rice and Diggle, Liang, and Zeger proposed methods for identifying representative curves from a large collection that display the form and extent of variation of the curves. Here we propose the use of tree-structured regression, as adapted to longitudinal data, for this purpose. Properties of each method are described. A re-examination of atmospheric ozone data, analyzed by Jones and Rice, is also presented.

A Simple Wavelet Approach to Nonparametric Regression from Recursive Partitioning Schemes

Rates of convergence for nonparametric regression estimators based on recursive partitioning schemes are derived. The central idea is to consider the tree-structured regression estimator as a wavelet estimator based on the orthogonal system of Haar functions. A locally adaptive data-driven smoothing method is proposed and its performance is studied.

A tree-structured piecewise linear adaptive filter

The authors propose and analyze a novel architecture for nonlinear adaptive filters. These nonlinear filters are piecewise linear filters obtained by arranging linear filters and thresholds in a tree structure. A training algorithm is used to adaptively update the filter coefficients and thresholds at the nodes of the tree, and to prune the tree. The resulting tree-structured piecewise linear adaptive filter inherits the robust estimation and fast adaptation of linear adaptive filters, along with the approximation and model-fitting properties of tree-structured regression models. A rigorous analysis of the training algorithm for the tree-structured filter is performed. Some techniques are developed for analyzing hierarchically organized stochastic gradient algorithms with fixed gains and nonstationary dependent data. Simulation results show the significant advantages of the tree-structured piecewise linear filter over linear and polynomial filters for adaptive echo cancellation. >

Prostate cancer treated by radiotherapy: a multivariate study

According to respective proportions of evolutive status groups, results of multivariate studies are difficult to interpret. Among the 1099 cases of local form of prostate cancer, treated by radiotherapy from 1975 to 1982 in 16 French Anticancer Institutes, we can observe two homogeneous status groups of patients: disease-free survivors (285 cases) and patients who died of prostate cancer (278 cases). These correspond to 51 % of the whole population. Among other things, they are comparable in size, for age at the beginning of radiotherapy and for delay between histologic diagnostic and radiotherapy. We chose to analyse them using multivariate analysis. To take survival into account, we used a Cox model and Kaplan-Meier curves; the group deceased of prostate cancer was further analyzed by a tree-structured regression method. The Cox model and the Kaplan-Meier curves confirmed two main explicative factors: Stage ( p<0.0001) and tumor grade ( p<0.001). Poorer evolution occurs in extracapsular forms and grade I has better survival than others. The tree-structured regression method indicates two other pejorative factors: hormonotherapy prior to radiotherapy and the presence of cardiovascular pathology. Though the pelvic dose does not appear to be a main explicative factor, it seems to improve survival and delay between radiotherapy and recurrence or metastasis in some categories of cases. Other factors such as tumor dose, age and delay between diagnosis and radiotherapy were not found to be significant. These results cannot be extended to the whole population for which they do not constitute a predictive study. We consider them as “baseline data”. In further studies, we would like to situate other evolving status groups in reference to the above results and finally study prostate cancer evolution by taking into account all evolving status groups according to their respective proportions and characteristics.

Variable rate vector quantization for medical image compression

Three techniques for variable-rate vector quantizer design are applied to medical images. The first two are extensions of an algorithm for optimal pruning in tree-structured classification and regression due to Breiman et al. The code design algorithms find subtrees of a given tree-structured vector quantizer (TSVQ), each one optimal in that it has the lowest average distortion of all subtrees of the TSVQ with the same or lesser average rate. Since the resulting subtrees have variable depth, natural variable-rate coders result. The third technique is a joint optimization of a vector quantizer and a noiseless variable-rate code. This technique is relatively complex but it has the potential to yield the highest performance of all three techniques.

Prevalence and risk factors of recurrent cervical pain syndrome in a working population.

The prevalence and risk factors of recurrent cervical pain syndrome (CPS) was studied in 990 workers examined while undergoing their systematic annual visit of occupational medicine. A prevalence rate of 10.2% was obtained. Using a tree-structured logistic regression analysis, a strong association between age, sex, previous head trauma extension of the neck and strenuous muscular activity during work, and the risk of recurrent CPS was obtained. Finally, an inverse association between strenuous muscular activity during leisure time and the risk of recurrent CPS was found.