## 3 Proposed Algorithms
### Variance Feedback  
The variance feedback strategy we proposed provides personalized drift detection strategies for different data streams based on the variance estimated by the variance estimation strategy. This strategy will provide different mean calculation methods and statistical testing methods for drift detection based on different variances. Improve the pertinence of drift detection, thereby improving the detection speed and accuracy of the drift detector, and at the same time improving the performance of the base classifier. The work of the variance feedback strategy is divided into two phases: the **Mean generation** and the **Statistical test selection**. **Phase 1. ** Mean generation  
First, in this stage, window instances will be weighted to reflect the difference between old and new data. We designed a weight adjustment strategy based on variance feedback. We believe that different variances represent different stability of data streams, so they should also be treated differently in weighting methods. In the field of concept drift detection, the purpose of weighting is to increase detection speed and reduce detection delay. Therefore, environments with lower variance can adopt a more relaxed weighting method than environments with higher variance, that is, the weight can be appropriately increased. This study uses a Sigmoid-like function to adjust the weight, and the weight adjustment strategy based on variance feedback can be expressed as Equation (10). \[diff\ =\ diff_{0}\cdot\left(1+\frac{1}{1+e^{\sigma^{2}}}\right) \tag{10}\]  
Among them, \(diff_{0}\) is the original weight factor. This formula reflects the design idea of using high weight in a more stable environment and low weight in a less stable environment. Secondly, we designed a mean adjustment strategy based on variance feedback, which has the same theoretical basis as the weight adjustment strategy. We believe that under lower variance conditions, the data stream is relatively stable and there are fewer influencing factors such as noise, so the window state is relatively stable. The drift detection  
Figure 4: Workflow of Variance estimation (Including the sampling process and the variance estimation process). strategy in this case can be more relaxed, the opposite is true in the case of higher variance. We used the variance \(\sigma^{2}\) to appropriately adjust the original observation mean \(p_{0}\) of the window, which is specifically expressed as Equation (11). \(\beta\) is the mean influence factor. \[p\ =\ \frac{p_{0}}{1-\beta/\left(1+e^{\sigma^{2}}\right)} \tag{11}\]  
**Phase 2. ** Statistical test selection  
In this phase, different statistical testing methods are provided to detect concept drift based on the variance of the data stream in a stable state. The selection strategy for statistical testing is based on the critical variance \(\sigma_{c}^{2}\) between the Bernstein bound and the Hoeffding bound, and it provides various concept drift statistical testing strategies for data streams through the relationship between the estimated variance value \(\sigma^{2}\) and \(\sigma_{c}^{2}\). **Discussion--**The above approach principle is as follows: The Bernstein bound can be represented by Equation (4), while the Hoeffding bound under the same conditions can be represented by Equation (12). \[\varepsilon_{H}\ =\ \sqrt{\frac{1}{2n}\ln\frac{1}{\delta}} \tag{12}\]  
Observing the Bernstein bound and the Hoeffding bound, we refer to the variance of the data stream when they are equal to the critical variance, which can be expressed as Equation (13). \[\sigma_{c}^{2}\ =\ \frac{1}{4}-\frac{1}{3}\varepsilon_{H} \tag{13}\]  
The critical variance \(\sigma_{c}^{2}\) reflects the applicability range of the Bernstein bound. When \(\sigma^{2}=\sigma_{c}^{2}\), the Bernstein bound is equal to the Hoeffding bound, meaning that the Bernstein test and the Hoeffding test have the same theoretical delay for drift detection.