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Advancements in Linear Discriminant Trees for Decision Tree Algorithms

This paper explores the construction of multivariate linear decision trees, focusing on combining linear discriminant analysis (LDA) with decision tree techniques. It presents a unique method involving binary splits and non-iterative training, highlighting various algorithms such as ID3, C4.5, CART, and neural trees. The methodology includes detailed procedures for class separation and entropy computation, utilizing datasets from the UCI repository for empirical validation. Results are evaluated based on accuracy, tree size, and learning time, demonstrating significant advancements in decision tree methodologies.

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Advancements in Linear Discriminant Trees for Decision Tree Algorithms

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  1. Linear Discriminant Trees Olcay Taner Yıldız, Ethem Alpaydın Department of Computer Engineering Boğaziçi University, Istanbul Turkey yildizol@yunus.cmpe.boun.edu.tr

  2. Decision Trees

  3. Decision Tree Algorithms • Univariate Algorithm • ID3, C4.5 (Quinlan 1986) • Multivariate Algorithms • CART (Breiman et al., 1984) • Neural Trees (Guo and Gelfand, 1992) • OC1 (Murthy, Kasif & Salzberg, 1994) • LMDT (Brodley and Utgoff, 1995)

  4. ID-LDA Tree Construction • Divide K classes in that node into two parts. (Outer Optimization) • Solve two class problem with LDA in that node. (Inner optimization) • For each of two child nodes repeat step 1 and step 2 recursively until each node has only one class in it.

  5. Class Separation by Exchange Method (Guo & Gelfand, 1992) • Select an initial partition of C into CL and CR, both containing K/2 classes • Train the discriminant to separate CL and CR. Compute the entropy E0 with the selected entropy formula • For each of the classes i in C1 ... Ck form the partitions CL(i) and CR(i) by changing the assignment of the class Ci in the partitions CL and CR • Train the neural network with the partitions CL(i) and CR(i). Compute the entropy Ei and the decrease in the entropy Ei=Ei-E0 • Let E* be the maximum of the impurity decreases over all possible i and i*be the i causing the largest decrease. If this impurity decrease is less than zero then exit else set CL=CL(i*), CR=CR(i*), and goto step 2

  6. Linear Discriminant Analysis

  7. PCA for Feature Extraction • Singular Matrix Problem Sw • Answer: Principal Component Analysis • Find most important k eigenvectors • Feature Extraction • PCA finds new k dimensions as linear combinations of d features. • Subset selection finds the best k dimensions discarding d-k features.

  8. Experiments • 20 data sets from UCI Repository are used • Three different criteria used • Accuracy • Tree Size • Learning Time • For comparison 52 cv F-Test is used. (Alpaydın, 1999)

  9. Results for Accuracy Results for Tree Size Results for Learning Time

  10. Conclusions • A novel method for constructing multivariate linear decision trees • Binary splits • No iterative training

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