TREE-BASED ENSEMBLE CLASSIFIERS FOR HIGH-DIMENSIONAL DATA

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1 Vol. 45, (2007)1 17 TREE-BASED ENSEMBLE CLASSIFIERS FOR HIGH-DIMENSIONAL DATA James J. Chen, Hojin Moon, and Songjoon Baek Division of Personalized Nutrition and Medicine National Center for Toxicological Research Food and Drug Administration Building a classification model from thousands of available predictor variables with a relatively small sample size presents challenges for most traditional classification algorithms. When the number of samples is much smaller than the number of predictors, there can be a multiplicity of good classification models. An ensemble classifier combines multiple single classifiers to improve classification accuracy. This paper overviews tree-based classifiers and compares the performance of the three ensemble classifiers: random forest (RF), classification by ensembles from random partitions (CERP), and adaptive boosting (AdaBoost), and three single tree algorithms are also evaluated, classification tree (CTree), classification rule with unbiased interaction selection and estimation (CRUISE), and quick, unbiased and efficient statistical tree (QUEST). The six tree-based classifiers are applied to five high-dimensional datasets. In all datasets, the three ensemble classifiers show much higher classification accuracies than the three single tree algorithms, with the exception of the AdaBoost ensemble classifier in one dataset. RF and CERP are comparable in terms of accuracy. The RF and CERP bagging classifiers show higher accuracies than the AdaBoost boosting classifier. For the three tree classifiers, QUEST generally shows higher accuracy than CTree and CRUISE. Key words and phrases: adaptive boosting (AdaBoost), bagging, boosting, classification by ensembles from random partitions (CERP), classification tree, random forest

2 Vol. 45, (2007)18 37 ASSESSING SOFTWARE RELIABILITY BY UNREVEALED PROPORTION ESTIMATION IN STRATIFIED SAMPLING Mark C. K. Yang 1, Anne Chao 2 and Y. C. Chen 3 1 Department of Statistics, University of Florida 2 Institute of Statistics, National Tsing Hua University 3 Chia-Nan University of Pharmacy and Science Suppose there is an unknown number of species in an area of interest. A sample is collected and all the animals are identified according to their species. The purpose is to estimate the proportion θ of the animals that belong to undiscovered species. Point and interval estimators for θ are derived when the area is stratified into K strata and observations are taken from each stratum. Use of proportional allocation is shown to be more efficient than simple random sampling. This model fits very well in software reliability estimation under beta testing, i.e., the software faults are detected by complaints from the users. Many difficult situations in software testing and reliability are overcome by this model. Key words and phrases: undiscovered species; proportional allocation; software reliability; software maintenance; beta testing. JEL classification: C13, C42, C88. Improving software reliability by testing is one of the most important components in software development. It has been estimated that in many projects, the efforts used

3 2 JAMES J. CHEN, HOJIN MOON, AND SONGJOON BAEK (RF), voting classifier, tree algorithm. JEL classification: C14; C45 Classification or class prediction (machine learning algorithm) is a widely used data mining technique in many areas of research and applications. Classification uses a supervised learning method where the classification algorithm learns from a training set, a set of predictors with known class labels, and establishes a prediction rule to classify new samples. Classification has been used to predict the activity or toxicological property of untested chemicals, for instance, to predict rodent carcinogenicity (Helma and Kramer, 2003), Salmonella mutagenicity (e.g., Rosenkranz and Cunningham, 2001), or estrogen receptor binding activity of chemicals (Chen et al., 2005, 2006) using structure-activity relationship models. Recently, classification models have been developed to discriminate different biologic or clinical phenotypes, or to predict the diagnostic category, prognostic stage of a patient, or treatment response in personalized medicine (e.g, Alon et al., 1999; Golub et al., 1999; Moon et al., 2007). Classification involving a large number of predictors presents a challenge to the development of accurate classifiers. Standard classification model building, such as the logistic regression and Fisher s linear discriminant analysis, has relied on an a priori collection of predictors and model selection that was limited to a few variables of interest. The classification tree (CTree) is known as binary recursive partitioning (Breiman et al., 1984). A tree is constructed iteratively in the direction of making data purer according to a splitting criterion until it is fully grown. The support vector machine (SVM) is a kernel-based machine-learning predictor (Vapnik, 1995). In a twoclass classification, an SVM classifier finds a hyperplane between the two classes that minimizes the error probability. Both CTree and SVM have incorporated important predictor variables into model building. These algorithms find a subset of predictors and evaluate its relevance for the classification; the classification rules are built from an optimal predictor subset.

4 Vol. 45, (2007)38 54 CHARACTERIZATIONS OF THE ORDER STATISTICS POINT PROCESS Wen-Jang Huang 1, Nan-Cheng Su 2 And Jyh-Cherng Su 3 1 National University of Kaohsiung 2 National Sun Yat-sen University 3 R.O.C. Military Academy Let A {A(t), t 0} be an order statistics point process, with E(A(t)) = m(t) being the mean value function of A(t), t 0. It is known that m(t) determines the distribution of the process A. In this work, we give some characterizations of m(t), by using certain relations between the conditional moments of the last jump time or current life of A at time t. It is interesting that some results are parallel to those characterizations of Poisson process as a renewal process. Finally, we present some extensions of the results about record values given in Abu-Youssef (2003). Key words and phrases: Characterization; nonhomogeneous Poisson process; order statistics; order statistics property; record values. JEL classification:. Let {A(t), t 0} with A(0) = 0, A(t) <, t 0, be a point process with right continuous sample paths having successive unit steps at times S 1, S 2,... The process {A(t), t 0} is said to have the order statistics property or called an order statistics point process if for every t > 0 and integer n 1, whenever P (A(t) = n) > 0, given

5 Vol. 45, (2007)55 73 AN EMPIRICAL APPLICATION OF MARKOV MODEL FOR THE TERM STRUCTURE OF CREDIT RISK SPREADS Alan T. Wang 1 and Wei-Chen Lee 2 1 Department of Accounting, National Cheng Kung University 2 Research Department, Tachung Bank The Markov models by Jarrow, Lando and Turnbull (JLT) (1997) and Kijima and Komoribayashi (KK) (1998) provide an important alternative for pricing financial instruments on credit risk and risk management. Albeit some extensions have already been developed, empirical analysis of JLT-KK model is less documented. This article implements the model of KK and reports empirical results. Using the credit spread term structure observed in the market, unconstrained risk premium adjustments for riskier bounds with longer maturities are shown to easily exceed the upper bounds when default-dependent recovery rates are used. Key words and phrases: credit risk, default probability, Markov. JEL classification: G12, G13, G32. The credit derivatives market has grown rapidly recently, and it s a booming business with $8.4 trillion outstanding contracts at the end of 2004, compared to $919 billion in 2001, according to the International Swaps and Derivatives Association. Credit derivatives are linked to the probability that the debts will be paid off, which is in turn linked to the credit status of the issuing firms and the value of the reference asset. For

6 Vol. 45, (2007)74 98 ACCOUNTING FOR TAIWAN GDP GROWTH: PARAMETRIC AND NONPARAMETRIC ESTIMATES Ling Sun 1 and Lilyan E. Fulginiti 2 1 Department of Accounting, Providence University 2 Department of Agricultural Economics, University of Nebraska-Lincoln The purpose of this paper is to study the impact of changes in prices of tradables on economic growth in a highly open economy, Taiwan. We do so by measuring productivity growth with both index number and parametric approaches, and identifying the sources of output growth using a methodology that allows the impacts of changes in the terms of trade to be accounted for. The results show that Taiwan s economic growth depends on inputs accumulation as well as technical progress with the terms-of-trade effect being negligible. Key words and phrases: productivity change; SUR (Seemingly Unrelated Regressions); nonparametric approach; stochastic approach; terms of trade. JEL classification: O30, C01, C14, C30. Productivity is defined as output per unit of input. The study of productivity is intimately related to the study of economic growth as productivity increases induce an increase in output in perpetuity while this might not be true of input use. In fact, estimated aggregate supply elasticities have been known to be very small. It is shifts in this aggregate supply due to innovations that has reverted Malthusian predictions and in fact allowed higher standard of living.

7 105 JCSA, March 2007, Vol.45, No.1, page On Teaching Statistics in High School Mark C. K. Yang Department of Statistics, University of Florida Teaching statistics in high school, the emphases should be on basic concepts, real examples, and misuses. Due to time and background constraints, most rigorous proofs are not possible. Hence, it cannot be in the mathematics curriculum. It should put in those areas where data analysis is greatly affected by noise, especially the noise due to personal differences, such as in behavioral science, social science, life science, education and medicine. Statistics can be taught as a key methodology in social science.

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