## 1 Introduction
[PERSON] pointed out that despite the promise of the MTS to classify imbalanced data, it lacks an effective threshold determination method, so they proposed a nonlinear optimization model based on a ROC curve to do this [17]. [PERSON] et al. used Fischer discriminant analysis to correct the MD and constructed a threshold optimization model; they then applied their improved MTS to the fault diagnosis of rolling bearings, whose balanced accuracy was experimentally shown to be significantly increased [18]. [PERSON] et al. improved the robustness of the MTS by integrating it with a bagging algorithm [19]. [PERSON] et al. constructed optimization models for feature selection and threshold determination [20]. All the above studies have improved the imbalanced data classification performance of the MTS, but most were proposed for binary classification problems. However, in practical applications such as disease detection, sentiment classification, and fault diagnosis, the data usually contain multiple classes, and their distribution is unbalanced. Hence, it is necessary to extend the MTS to imbalanced multi-classification problems. In addition, most studies optimize the MTS in stages, which may not be conducive to its further enhancement. To further broaden the application field of MTS, improve its theoretical foundation, enhance its performance of imbalanced data multi-classification, and enrich the methods of imbalanced data multi-classification, we propose a more powerful imbalanced multi-classification method called optimized binary tree MTS (optimized BT-MTS). First, considering that the MS of the MTS is the only population, referring to the research of [PERSON] et al. on BT-MTS [21], the optimized BT-MTS combines the MTS with a binary tree to realize its multi-classification application in a one-versus-all (OVA) manner. However, the approach increases the imbalance ratio of the data. Hence, MS construction, feature selection, and threshold determination are incorporated into a unified classification framework for joint optimization. The unified classification framework enables the class-selection, feature-selection, and classification process of the MS to simultaneously update the sample separability information, feature representation, and classifier information to obtain the optimal solution for the whole learning process. The unified classification framework optimizes the construction of the MS, feature selection, and threshold determination of the MTS based on the principles of maximizing the class separability, signal-to-noise ratio of normal and abnormal samples, dimensionality reduction, and minimizing the misclassification cost. Hybrid particle swarm optimization (HPSO) is used to solve what is typical mixed integer nonlinear programming (MINLP). To verify the effectiveness of the algorithm,it was compared with BT-MTS, random forest (RF), fast correlation-based filter solution support vector machine (FCBF-SVM), synthetic minority oversampling K-nearest neighbor (SMOTE-KNN), and a backpropagation neural network (BPNN). Experimental results on several public datasets showed that the optimized BT-MTS performed better in the classification of unbalanced data as it had higher scores for macro-F1, kappa coefficient, and balanced accuracy, and lower total misclassification cost. The innovations of the optimized BT-MTS are as follows: (1) a multi-classification application of the MTS is implemented and optimized through an OVA architecture, and a multi-classification method for unbalanced data based on the MTS is explored. In addition, it is the first time MTS is used to solve the imbalanced data multi-classification problem; (2) a unified classification framework is proposed to jointly optimize the MS construction, feature selection, and classifier learning through an iterative approach, avoiding suboptimal solutions that may arise from phased optimization.