Decision tree classifiers are a widely used tool in data stream mining. The use of confidence intervals to estimate the gain associated with each split leads to very effective methods, like the popular Hoeffding tree algorithm. From a statistical viewpoint, the analysis of decision tree classifiers in a streaming setting requires knowing when enough new information has been collected to justify splitting a leaf. Although some of the issues in the statistical analysis of Hoeffding trees have been already clarified, a general and rigorous study of confidence intervals for splitting criteria is missing. We fill this gap by deriving accurate confidence intervals to estimate the splitting gain in decision tree learning with respect to three criteria: entropy, Gini index, and a third index proposed by Kearns and Mansour. Our confidence intervals depend in a more detailed way on the tree parameters. We also extend our confidence analysis to a selective sampling setting, in which the decision tree learner adaptively decides which labels to query in the stream. We furnish theoretical guarantee bounding the probability that the classification is non-optimal learning the decision tree via our selective sampling strategy. Experiments on real and synthetic data in a streaming setting show that our trees are indeed more accurate than trees with the same number of leaves generated by other techniques and our active learning module permits to save labeling cost. In addition, comparing our labeling strategy with recent methods, we show that our approach is more robust and consistent respect all the other techniques applied to incremental decision trees.
In this work, we significantly simplify the approach of @cite_6 and extend it to a third splitting criterion. Moreover, we also solve the bias problem, controlling the deviations of @math from the real quantity of interest (i.e., @math rather than @math ). Moreover, unlike @cite_29 and @cite_27 , our bounds apply to the standard splitting criteria. Our analysis shows that the confidence intervals associated with the choice of a suboptimal split not only depend on the number of leaf examples @math ---as in bounds ) and )--- but also on other problem dependent parameters, as the dimension of the feature space, the depth of the leaves, and the overall number of examples seen so far by the algorithm. As revealed by the experiments in , this allows a more cautious and accurate splitting in complex problems. Furthermore, we point out that our technique can be easily applied to all extensions of VFDT (see ) yielding similar improvements, as these extensions all share the same Hoeffding-based confidence analysis as the Hoeffding tree algorithm.
@cite_27 @cite_29 @cite_6
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Many stream classification algorithms use the Hoeffding Inequality to identify the best split attribute during tree induction. We show that the prerequisites of the Inequality are violated by these algorithms, and we propose corrective steps. The new stream classification core, correctedVFDT , satisfies the prerequisites of the Hoeffding Inequality and thus provides the expected performance guarantees. The goal of our work is not to improve accuracy, but to guarantee a reliable and interpretable error bound. Nonetheless, we show that our solution achieves lower error rates regarding split attributes and sooner split decisions while maintaining a similar level of accuracy. A novel algorithm based on random forests with surrogate splits is proposed to address the classification problem of incomplete data without imputation.The algorithm allows each tree to cast a vote even the voting process is interrupted by missing attributes.Experimental results on various acknowledged datasets show that the proposed method is robust and efficient. Random forests (RF) is known as an efficient algorithm in classification, however it depends on the integrity of datasets. Conventional methods in dealing with missing values usually employ estimation and imputation approaches whose efficiency is tied to the assumptions of data features. Recently, algorithm of surrogate decisions in RF was developed and this paper proposes a random forests algorithm with modified surrogate splits (Adjusted Weight Voting Random Forest, AWVRF) which is able to address the incomplete data without imputation.Differing from the present surrogate method, in AWVRF algorithm, when the primary splitting attribute and the surrogate attributes of an internal node are all missing, the undergoing instance is allowed to exit at the current node with a vote. Then the weight of the vote is adjusted by the strength of the involved attributes and the final decision is made by weighted voting. AWVRF does not comprise imputation step, thus it is independent of data features.AWVRF is compared with the methods of mean imputation, LeoFill, knnimpute, BPCAfill and conventional RF with surrogate decisions (surrRF) using 50 times repeated 5-fold cross validation on 10 acknowledged datasets. In a total of 22 experiment settings, the method of AWVRF harvests the highest accuracy in 14 settings and the largest AUC in 7 settings, exhibiting its superiority over other methods. Compared with surrRF, AWVRF is significantly more efficient and remain good discrimination of prediction. Experimental results show that the present AWVRF algorithm can successfully handle the classification task for incomplete data. In mining data streams the most popular tool is the Hoeffding tree algorithm. It uses the Hoeffding's bound to determine the smallest number of examples needed at a node to select a splitting attribute. In the literature the same Hoeffding's bound was used for any evaluation function (heuristic measure), e.g., information gain or Gini index. In this paper, it is shown that the Hoeffding's inequality is not appropriate to solve the underlying problem. We prove two theorems presenting the McDiarmid's bound for both the information gain, used in ID3 algorithm, and for Gini index, used in Classification and Regression Trees (CART) algorithm. The results of the paper guarantee that a decision tree learning system, applied to data streams and based on the McDiarmid's bound, has the property that its output is nearly identical to that of a conventional learner. The results of the paper have a great impact on the state of the art of mining data streams and various developed so far methods and algorithms should be reconsidered. We prove the approximation ratio 8 5 for the metric s,t -path-TSP, and more generally for shortest connected T-joins. The algorithm that achieves this ratio is the simple "Best of Many" version of Christofides' algorithm (1976), suggested by An, Kleinberg and Shmoys (2012), which consists in determining the best Christofides s,t -tour out of those constructed from a family @math of trees having a convex combination dominated by an optimal solution x* of the Held-Karp relaxation. They give the approximation guarantee @math for such an s,t -tour, which is the first improvement after the 5 3 guarantee of Hoogeveen's Christofides type algorithm (1991). Cheriyan, Friggstad and Gao (2012) extended this result to a 13 8-approximation of shortest connected T-joins, for |T|≥4. The ratio 8 5 is proved by simplifying and improving the approach of An, Kleinberg and Shmoys that consists in completing x* 2 in order to dominate the cost of "parity correction" for spanning trees. We partition the edge-set of each spanning tree in @math into an s,t -path (or more generally, into a T-join) and its complement, which induces a decomposition of x*. This decomposition can be refined and then efficiently used to complete x* 2 without using linear programming or particular properties of T, but by adding to each cut deficient for x* 2 an individually tailored explicitly given vector, inherent in x*. A simple example shows that the Best of Many Christofides algorithm may not find a shorter s,t -tour than 3 2 times the incidentally common optima of the problem and of its fractional relaxation.