## 3 Proposed Algorithms
### Variance Feedback
When \(\sigma^{2}<\sigma_{c}^{2}\), the Bernstein bound will be smaller than the Hoeffding bound [19], resulting in a lower drift detection delay compared to the Hoeffding test. Additionally, as the variance decreases further, the drift detection delay of the Bernstein test will also decrease further. Therefore, in environments where \(\sigma^{2}\leq\sigma_{c}^{2}\), the Bernstein test shows a more pronounced advantage in detection speed. Reference [5] suggests that for low variance distributions, the Hoeffding bound overestimates the probability of large deviations. Therefore, using Bernstein testing in such cases theoretically provides a faster drift detection speed. Conversely, when \(\sigma^{2}>\sigma_{c}^{2}\), the Bernstein bound will be greater than the Hoeffding bound and the Bernstein test will lag behind the Hoeffding test in terms of detection speed. Moreover, as the variance increases, the disadvantage of the Bernstein test will become more pronounced. So, in this case, [PERSON]'s test theoretically becomes suboptimal for drift detection. Based on the above analysis, the statistical test selection strategy can be expressed as Equation (14). \[\left\{ \begin{array}{l}if:\sigma^{2}\leq\sigma_{c}^{2}\rightarrow\text{ Bernstein Test}\\ if:\sigma^{2}>\sigma_{c}^{2}\rightarrow\text{Other Statistical Test}\end{array} \right. \tag{14}\]  
Figure 5 illustrates the overall workflow of the variance feedback strategy. First, in Phase 1, the strategy uses the estimated variance \(\sigma^{2}\) to generate the instance weight factor _diff_. Then, the initial mean \(p_{0}\) of the window instances is generated. Further, the final mean \(p\) is generated based on the estimated variance and \(p_{0}\), and the maximum mean \(p_{max}\) is updated. In Phase 2, the estimated variance \(\sigma^{2}\) is used to determine the statistical test method for drift detection, thus completing the drift detection process. From Equation (14), it can be inferred that when \(\sigma^{2}>\sigma_{c}^{2}\), the statistical test selection strategy will opt for other more compact statistical testing methods for drift detection. However, not all testing methods are suitable; this is due to considerations regarding detection speed. Reviewing the relationship between the Bernstein bound and the Hoeffding bound, the main reason for rejecting Bernstein testing under the condition of \(\sigma^{2}>\sigma_{c}^{2}\) is that in this scenario, the Bernstein bound is larger, theoretically decreasing the drift detection speed. Therefore, alternative statistical testing methods chosen in this case should have faster detection speeds, namely, lower testing bounds. Based on the aforementioned requirements, this section will introduce some statistical tests suitable for environments where \(\sigma^{2}>\sigma_{c}^{2}\). **Statistical Test 1. ** Hoeffding's Test for Drift Detection  
The Hoeffding test is a commonly used statistical testing method in the field of concept drift detection. This test relies on [PERSON]'s inequality. **Theorem 2**. : _Hoeffding's inequality--Let \(X_{1}\), \(X_{2}\),..., \(X_{n}\) be \(n\) independent random variables located in the interval [0,1], with the variance is \(\sigma^{2}\) and any \(\epsilon_{H}>0\). Then,_  
\[\Pr\left(\left|\frac{1}{n}\sum_{i=1}^{n}X_{i}-E[X]\right|>\epsilon_{H}\right) \leq 2\exp\left(-2n\epsilon_{H}^{2}\right) \tag{15}\]  
Therefore, when a significance confidence level \(\delta\) is given, the significance threshold \(\epsilon_{H}\) required for the Hoeffding test can be expressed as Equation (12). Reviewing Equations (4) and (12), it can be observed that when \(\sigma^{2}>\sigma_{c}^{2}\), \(\epsilon_{H}<\epsilon_{B}\). Therefore, the Hoeffding test is a statistical testing method suitable for environments where \(\sigma^{2}>\sigma_{c}^{2}\), possessing drift detection performance that is at least superior to that of the Bernstein test. **Statistical Test 2.