## 3 Technical Design of GSFR-BLS
### Inference Result Smoothing for Robust Classification  
To further enhance the robustness of the classification process, a two-stage smoothing strategy is applied to the raw classification results from the BLS. The raw classification output \(o(t)\) from the BLS, representing the predicted probabilities for each class, is smoothedusing a combination of sliding window smoothing and EMA to reduce short-term fluctuations and maintain sensitivity to long-term trends.  
The first step is to apply a sliding window over the BLS classification results to reduce short-term fluctuations. This involves averaging the classification outputs over a window of consecutive frames. The sliding window-smoothed classification result is calculated as follows:  
\[o^{\text{smoothed}}(t)=\frac{1}{w}\sum_{j=t-w+1}^{t}o(j) \tag{13}\]  
where \(o(j)\) represents the classification output at frame \(j\), \(w\) is the size of the sliding window, and \(o^{\text{smoothed}}(t)\) is the smoothed classification output at frame \(t\).  
After applying sliding window smoothing, the classification results are further smoothed using EMA. This step ensures that the classification output responds to long-term trends while reducing jitter. The EMA-smoothed classification result is computed as  
\[o^{\text{EMA}}(t)=\gamma\cdot o^{\text{smoothed}}(t)+(1-\gamma)\cdot o^{ \text{EMA}}(t-1) \tag{14}\]  
where \(o^{\text{EMA}}(t)\) is the exponentially smoothed classification result at frame \(t\), \(\gamma\) is the smoothing factor, and \(o^{\text{EMA}}(t-1)\) is the EMA result from the previous frame.  
The final classification result after the dual smoothing process is given by  
\[o^{\text{final}}(t)=o^{\text{EMA}}(t). \tag{15}\]  
This final smoothed classification output \(o^{\text{final}}(t)\) reduces false positives and stabilizes the BLS predictions, making the classification more robust in real-world applications. This dual-smoothing approach effectively reduces noise in the classification outputs, improving robustness against false positives.  
Most parameters in Equations (1)-(15) are determined dynamically during data processing and model training, reducing the need for manual tuning and enhancing robustness across diverse scenarios. However, certain key parameters require explicit initialization, including the time window (\(T\)), dynamic threshold multiplier (\(\beta\)), regularization strength (\(\lambda\)), sliding window size (\(w\)), and smoothing factor (\(\gamma\)). The time window is set to 60 frames (approximately 2 s) as suggested by [29], aligning with typical human action durations in behavior recognition tasks. The dynamic threshold multiplier (\(\beta\)) is set to 2.0 based on the 3\(\sigma\) principle [30], effectively filtering noise while retaining relevant data. The regularization strength (\(\lambda\)) is set to 0.01 following [31] to prevent overfitting, while the sliding window size (\(w\)) of 10 frames and smoothing factor (\(\gamma\)) of 0.8 are chosen based on [32,33], respectively, to ensure stable and noise-resistant temporal feature extraction. These parameter settings balance computational efficiency and accuracy, providing a reliable foundation for our approach.