## 3 Proposed Algorithms
### Variance Feedback Drift Detection Method  
We devised and implemented VFDDM. This method utilizes a variance feedback mechanism. Firstly, our variance estimation strategy is employed to estimate the variance under the stable state of the data stream. Once the variance estimation is obtained, our variance feedback strategy is utilized to select a personalized drift detection scheme for the data stream. This includes setting up the window mean calculation method and selecting the statistical testing method. The overall schematic diagram of this method is illustrated in Figure 6.  
Then we will illustrate the workflow of the VFDDM using the drift detector VFDDM_H based on the Hoeffding test as an example. The VFDDM_H employs a single sliding window mechanism [21], with a window size of \(n\). The workflow of VFDDM_H is as follows:  
**Step. 1. ** The prediction results of the data stream instances (1 for correct, 0 for incorrect) will sequentially enter the window. **Step. 2. ** When the window is full, VFDDM uses the estimated variance \(\sigma^{2}\) to generate weight _diff_ and mean \(p\) for the current window instances and updates the historical maximum mean \(p_{max}\), as shown in Equation (22) [6]. \[\text{if }p_{max}<p\Rightarrow p_{max}\ =\ p \tag{22}\]  
**Step. 3. ** Based on the variance \(\sigma^{2}\), the statistical test method is selected for the current situation. When \(\sigma^{2}\leq\sigma_{c}^{2}\), the Bernstein test is employed; when \(\sigma^{2}>\sigma_{c}^{2}\), the Hoeffding test is used. This process is illustrated in Equation (23). \[\left\{ \begin{aligned} & if:\sigma^{2}\leq\sigma_{c}^{2} \to p_{max}-p>\varepsilon_{B}\\ & if:\sigma^{2}>\sigma_{c}^{2}\to p_{max}-p>\varepsilon_{H} \end{aligned} \right.\to Drift:True \tag{23}\]  
Meanwhile, based on \(p\), \(p_{max}\), and Equation (7), it is determined whether the current state meets the variance sampling conditions. If it does, the latest instance prediction result is added to the variance sample pool, and the estimated variance \(\sigma^{2}\) will be updated. **Step. 4. ** When the next instance prediction result enters the window, repeat **Step 1** to **Step 3**. In addition to VFDDM_H, we also implemented two other drift detectors: VFDDM_M based on the McDiarmid test and VFDDM_K based on the Kolmogorov test. For ease of exposition, we collectively refer to these three drift detectors, VFDDM_H, VFDDM_M, and VFDDM_K, as VFDDMs. Algorithm 1 provides a pseudocode summary of VFDDM. The inputs to Algorithm 1 include the data stream \(S\), the window \(win\), and the variance sample pool \(vp\), along with  
Figure 6: Workflow of VFDDM (Including variance estimation, variance feedback, and drift detection stages). two parameters--the confidence level \(\delta\) and the initial weighting factor \(\mathit{diff}_{0}\). Lines 2-3 describe the sliding window insertion and eviction operations when a new instance arrives. Lines 4-5 describe the operation of the variance feedback strategy, including weight calculation (Equation (10)) and mean calculation (Equation (11)). Line 6 represents the update process for the maximum mean of the window (Equation (22)). Lines 8-10 describe the mean generation process in the variance estimation strategy. When the current condition satisfies the variance sample collection condition (Equation (7)), the most recent predicted instance is added to the sample pool \(\mathit{vp}\). Additionally, the variance \(\sigma^{2}\) is updated according to Equation (9). Lines 11-18 describe the process of selecting the statistical test method in the variance feedback strategy. First, the critical variance \(\sigma_{c}^{2}\) is calculated based on Equation (13) (Line 11). Then a decision is made: when the variance is small (\(\sigma^{2}\leq\sigma_{c}^{2}\)), VFDDM adopts the Bernstein test; when the variance is large (\(\sigma^{2}>\sigma_{c}^{2}\)), VFDDM adopts other test methods.