## 1 Introduction  
Learning from non-stationary data streams remains a focal point in the research of stream data mining, and significant progress has been made in obtaining useful models from massive and rapidly generated data. Traditional static batch processing methods are no longer suitable for non-static learning environments, which pose new requirements for data stream processing [1]. New instances may arrive one by one or in batches, and the incoming instances must be classified within a limited time frame with limited resources. Classification algorithms should consider the issue of class imbalance in data streams. Data streams generated in real-world applications often involve multiple classes and imbalanced data ratios, and the ratio of classes may change as the stream continues, further deteriorating the learning ability of classifiers [2]. The class with a larger number of instances is referred to as the majority class, while the class with a smaller number of instances is referred to as the minority class that usually contains the information of interest. In data stream processing, the phenomenon of class imbalance becomes crucial because it occurs in various fields, such as network intrusion detection (network access traffic continuously arrives, and intrusion or attack traffic belongs to the minority class) [3], credit card fraud detection (credit card transaction records continuously generate, and fraudulent transactions on unsafe websites belong to the minority class) [4], disease diagnosis (in a large amount of disease data, patients with a specific disease belong to the minority class) [5], etc. Moreover, in multi-class imbalanced data streams, there may be constant changes in concepts, resulting in concept drift, which can significantly reduce the classification performance of classifiers [6]. Currently, there are relatively few methods for solving class imbalance problems containing concept drift [7]. Moreover, most of the existing methods are used to solve binary imbalance problems, which deal with only one minority class and one majority class. In multi-class environments, it is more challenging to deal with imbalance and concept drift problems. Not only is it necessary to deal with multiple minority classes, multiple majority classes, and concept drift simultaneously, but the data stream may be extremely imbalanced, and even the ratio of classes may change over time. Among the currently known methods, the Metacognitive Online Sequential Extreme Learning Machine (MOS-ELM) [8] is the first method that can handle multi-class imbalanced concept drift data streams with a variable class ratio. The Adaptive Random Forest Resampling algorithm (ARF\({}_{\text{Re}}\)) [9] can be used to solve the mixed concept drift problem with multi-class imbalance and variable class ratio. However, current algorithms have several shortcomings. On the one hand, classifying in extremely imbalanced data streams is a challenging task, as it further degrades the classifier's ability to recognize minority classes. On the other hand, existing sampling methods do not take into account the original distribution of the data during the sampling process. In this paper, we propose a Hybrid Sampling and Dynamic Weighting-based classification method for Multi-class Imbalanced data streams (HSDW-MI) to address the issues of imbalance, concept drift, and variable class ratio in data streams. The main contributions of this work are shown below:  
1. A novel hybrid sampling method is proposed to solve the problem of imbalance and variable class ratio in the data streams. In the hybrid sampling process, each class is clustered by adaptive spectral clustering and then sampled based on the clusters from each class. Meanwhile, a sample storage pool is designed to store samples with a high safety factor in the data chunk to solve the problems caused by extremely imbalanced and variable class ratio data streams.