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Date Received: 26 March 2022 / Date Revision: 20 April 2022 / Date Accepted: 23 April 2022 / Date Published: 26 April 2022

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As a well-known machine learning method, decision tree is widely used in the field of classification and recognition. In this paper, we propose a new confidence entropy-based decision tree method for the uncertainty of labels handled by the belief function, and generalize it to random forests. With Gaussian mixture models, this tree approach can directly handle continuous attribute values ​​without discretization preprocessing. Specifically, tree methods employ belief entropy, an uncertainty measure based on latent belief assignments, as a novel attribute selection tool. To improve classification performance, we construct an underlying tree-based random forest and discuss different predictive combination strategies. Some numerical experiments on the UCI machine learning dataset show that the proposed method achieves good classification accuracy in various situations, especially on data with high uncertainty.

Decision trees are widely used because of their good learning ability and ease of understanding. In some real cases, the cases may be little known due to factors such as randomness, incomplete data, and even uncertain subjective opinions of experts; however, traditional decision trees can only deal with specific patterns with precise data. Cases with imperfect observations are often ignored or replaced by accurate cases, although they may contain useful information [1], which may result in a loss of accuracy.

Over the past few decades, there have been many attempts to build trees from incomplete data. Probability trees [2, 3] are proposed based on probability theory and are usually an intuitive first tool for modeling uncertainty in practice; however, it turns out that probability is not always sufficient to represent the uncertainty of data [4, 5] (often called epistemic uncertainty). Various methods have been proposed to overcome this shortcoming, including: fuzzy decision trees [6, 7], probabilistic decision trees [8] and uncertain decision trees [9, 10]. In addition to the above approaches, it has been shown that a more general framework called belief function theory [11, 12] (also known as evidence theory or Dempster-Shafer theory) can model all types of knowledge. In recent years, the process of embedding belief functions into decision tree techniques has been extensively studied [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. In particular, among these methods, some trees [17, 18, 19] estimate parameters by maximizing the evidence likelihood function

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However, existing incomplete data methods do not fully consider continuous attributes. These proposals deal with uncertain data modeled by confidence functions and build trees by extending traditional decision tree methods. Imitation and deformation selection use existing methods to deal with continuous attribute values ​​through discretization, which poses the problem of losing training data details. For example, the information gain ratio, a metric for attribute selection in C4.5, is reformulated to fit evidence labels for the training set in C4.5 belief trees [19], where continuous-valued attributes are divided into four equally wide pre-learned intervals. This question leads to the purpose of this paper: to learn from uncertain data with joint attribute values ​​without preprocessing.

To achieve this, we first fit the training data for each attribute to a Gaussian mixture model (GMM), which consists of a model-wise normal distribution corresponding to the class labels, using

Algorithm. This step differs significantly from other decision trees, confirming the ability to handle lesser-known labels and raw attribute values ​​(discrete or continuous). Based on these GMM models, we generate a base belief assignment (BBA) and compute belief entropy [28]. The attribute with the smallest mean entropy that best distinguishes the classes will be selected as the partitioning attribute. The following decision tree induction steps are well designed and logical. To the best of our knowledge, this paper is the first to introduce GMM models and belief entropy to decision trees with evidence data.

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Another part of our proposal is to employ an ensemble approach for our entropic belief trees. Inspired by the idea of ​​building bagged trees based on random sampling [29], we further choose a more efficient and popular technique – Random Forest [30]. Under the belief function framework, the underlying tree will produce exact or batch (BBA modeling) label predictions, while traditional random forests can only cluster exact labels. Therefore, a new way of summarizing the predictions of the underlying trees is proposed, directly clustering batches of labels instead of voting on exact labels. The quality of this pooling preserves as much uncertainty as possible from the data, helping to produce more reasonable forecasts. The new combination method will be discussed later and compared to the traditional majority voting method.

We note that we presented our earlier work in a shorter conference paper [31]. Compared to our original conference paper, this paper improves on single-tree attribute selection and splitting strategies, and introduces ensemble learning into our tree approach.

Algorithms and Belief Entropy. Section 3 details the inductive process of the belief entropy method and proposes three different case prediction techniques. In Section 4, we introduce how single-belief entropy trees can be extended to random forests and discuss different predictive combination strategies. In Section 5, we detail our experiments on some classic UCI machine learning datasets to compare the classification accuracy of the proposed trees and random forests. Finally, Section 6 concludes with conclusions.

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Take its value among K classes. Each attribute has discrete finite values ​​or receives a value continuously within an interval. Classification learning is based on the entire training set of exact data, which contains labels labeled

However, there is incomplete knowledge of the input (feature vectors) and output (classification labels) in practical applications. Traditionally, imperfect knowledge is often modeled by probability theory, which has been found to be problematic in various contexts. Therefore, in this paper, we use confidence functions to model uncertainty. We usually think of attribute values ​​as exact, which can be continuous or discrete, while only output labels are indeterminate.

Decision trees [32] are considered to be one of the most effective machine learning methods and are widely used in practice to solve classification and regression problems. Success depends heavily on constructs that are understandable to both humans and computers. In general, decision trees are derived top-down from the training set T by recursively repeating the following steps:

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Several decision tree algorithms have been proposed for different attribute selection methods, such as ID3 [32], C4.5 [33] and CART [34]. Among these trees, ID3 and C4.5 choose entropy as an informative metric to compute and evaluate the quality of nodes divided by a given attribute.

Is the cardinality of the set of instances belonging to parent node and child node i.

A limitation of information retrieval is that the attribute with the largest value will be promoted the most [33], leading to

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It is not difficult to see that formula (2) is actually Shannon entropy. However, in this paper, an attribute selection method based on belief entropy [28] instead of Shannon entropy is newly designed for the characteristics of evidence data described by the belief function framework.

In order to improve the classification accuracy and generalization ability of machine learning, an ensemble model method is introduced into the learning process. An important branch of ensemble methods is called

, which simultaneously builds multiple base models learned from different training sets generated from raw data sampled from the bootstrap system. Based on bagged decision trees, Random Forest (RF) [30] not only randomly selects training instances, but also introduces randomness into attribute selection. Specifically, a traditional decision tree chooses the best split attribute among all D attributes; a random

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