## 3 Proposed Solution to Predict the AoA
### System Model and Mathematical Formulation
#### 3.3.4 Predictive AoA Models (DE-ACSW-R)  
**Weighted and Clustered Linear Regression for Predictive AoA Estimation**  
Linear Regression (LR) is a fundamental technique for forecasting a signal's future AoA. In the proposed DE-ACSW-R framework, LR is employed at the O-RU level due to its computational efficiency and suitability for real-time low-latency prediction in O-RAN Split 7-2x architecture. To improve performance in realistic high-mobility scenarios, our model extends classical LR with both temporal weighting and clustered sliding window logic, referred to as Adaptive DE-ACSW-R.  
Let \(\theta(t)\) denote the estimated AoA at a discrete time index, \(t\), and let the O-RU maintain a sliding window, \(D_{t}\), of the \(W\) most recent AoA samples:  
\[D_{t}=\{(t_{i},\theta_{i})\}_{i=t-W+1}^{t} \tag{6}\]  
where \(t_{i}\) is the discrete time index (e.g., snapshot or frame number), \(\theta_{i}=(t_{i})\) is the estimated AoA at the time, and \(t_{i}\)\(W\) is the prediction window size. We fit a simple linear model of the following form:  
\[\hat{\theta}(t+\Delta)=at+b \tag{7}\]  
where a is the slope representing an angular rate of change (AoA velocity), b is the intercept, and \(\Delta\) is the forward prediction step (typically \(\Delta=1\) for next-slot prediction). The parameters a and b are computed via ordinary least squares (OLS) minimization:  
\[(a,b)=arg\min_{a,b}\sum_{i=t-W+1}^{t}(\theta_{i}-(at_{i}+b))^{2} \tag{8}\]  
This minimisation yields closed-form solutions:  
\[a=\frac{W\sum t_{i}\theta_{i}-\sum t_{i}\sum\theta_{i}}{W\sum t_{i}^{2}-(\sum t _{i})^{2}},\quad a=\frac{\sum\theta_{i}-a\sum t_{i}}{W} \tag{9}\]  
Once the model is trained, the predicted AoA for future time \(t+\Delta\) is given by the following:  
\[\hat{\theta}_{LR}(t+\Delta)=a(t+\Delta)+b \tag{10}\]  
Depending on the system confidence level, this prediction is then passed to the enhanced MUSIC estimator or is directly used for precoding. **Weighted Linear Regression (WLR) Extension**  
To improve robustness in rapidly changing mobility scenarios, we implement a WLR variant. This model gives more importance to recent AoA values by assigning temporal weights. \[(a,b)=arg\min_{a,b}\sum_{i=t-W+1}^{t}w_{i}(\theta_{i}-(at_{i}+b)) \tag{11}\]  
where \(w_{i}\) is a monotonically increasing function over time (e.g., exponential weights):  
\[w_{i}=\exp(\gamma(t_{i}-t_{W}))\,,\;\gamma>0 \tag{12}\]  
This ensures that recent AoAs are prioritised, allowing for faster adaptation to mobility-induced angular shifts while maintaining mathematical tractability. **Adaptive Clustered Sliding Window Regression (Jump Detection)**  
To further improve prediction under abrupt mobility changes, we introduce a jump-detection and clustering logic:  
* If \(|\theta_{k+1}-\theta_{k}|>\)_jump threshold_ for some k in the window, then only the samples after the last jump (i.e., from \(t_{k+1}\)_to_\(t\)) are used in the weighted regression. * If no jump is detected, the entire window is used. The prediction is then as follows: \[\theta_{ACSW-R}(t+\Delta)=a^{*}(t+\Delta)+b^{*}\] (13) where \(a,\;b\) are obtained by weighted regression on the \"clustered\" (post-jump) segment. **Linear Regression-based AoA prediction relies on the following assumptions:**  
* Short-term linear trend: The AoA trajectory over the prediction window is approximately linear, valid for low-speed UEs or over short time scales. * Stationarity within window: The angular velocity \(\frac{\mathrm{d}\theta}{\mathrm{d}t}\) remains roughly constant within the window, W.
* High-quality historical AoAs: The recent AoA estimates \(\theta_{i}\) are accurate and not corrupted by multipath-induced ambiguity or outliers. Given its simplicity and efficiency, LR is ideally suited for execution at the O-RU layer, where real-time responsiveness and minimal computational overhead are critical. When the Mean Squared Error (MSE) of the LR model falls below a predefined threshold (e.g., \(MSE_{LR}<1\)), the prediction is accepted.