## 3 Proposed Framework
### Online Testing Phase  
In this paper, we extend our previous work by presenting the results of an advanced ego vehicle localization methodology based on combined dictionaries. Our earlier approach [35] used individual interaction dictionaries to localize the ego vehicle through distinct, separate prediction models. In contrast, the methodology described herein integrates multiple track interaction dictionaries into a unified framework, significantly enhancing the robustness of localization. During the online testing phase, we employ a LiDAR-based dataset consisting of track observations captured in realistic driving scenarios. The localization process leverages \begin{table}
\begin{tabular}{c p{142.3 pt}} \hline \hline
**Step** & **Description** \\ \hline \multirow{2}{*}{1} & Temporal Alignment: Synchronize LiDAR and odometry timestamps to associate ego and track observations frame-wise. \\ \hline
2 & Track Association: Identify active TrackIDs using the JPDA filter at each frame. \\ \hline \multirow{2}{*}{3} & Cluster Assignment: Assign ego and track positions to their respective GNG clusters to extract cluster indices. \\ \hline \multirow{2}{*}{4} & Interaction Labeling: If ego and track positions fall within a defined spatial proximity, label the interaction as \(L=1\); otherwise \(L=0\). \\ \hline \multirow{2}{*}{5} & Dictionary Population: Store the interaction entry as \(I=\{t,C_{\text{ego}},C_{\text{track}},L,x,y\}\), where \((x,y)\) is the ego vehicle position. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Steps followed to construct the interaction dictionary between the ego vehicle and the observed tracks. Each step transforms raw frame-level data into structured interaction entries. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Module** & **Parameter** & **Description and Value** \\ \hline \multirow{3}{*}{JPDA Tracker} & Gating threshold & 0.8 (based on Mahalanobis distance) \\  & Confirmation count & 3 consecutive detections required \\  & Missed detection tolerance & Up to 5 missed updates allowed \\ \hline \multirow{6}{*}{GNG Clustering} & Maximum number of nodes & 100 (adaptive per trajectory length) \\  & Learning rate (winner) \(\epsilon_{b}\) & 0.05 \\ \cline{1-1}  & Learning rate (neighbor) \(\epsilon_{n}\) & 0.0006 \\ \cline{1-1}  & Maximum edge age & 50 iterations \\ \cline{1-1}  & Node insertion interval & Every 100 steps \\ \cline{1-1}  & Error decay factor \(\beta\) & 0.0005 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameter settings used in the JPDA tracker and GNG clustering algorithm. All values were empirically tuned using the training dataset. vocabulary and interaction dictionaries learned during the offline training phase. The proposed framework comprises distinct steps: track-cluster matching, particle initialization, prediction, and update phases. Importantly, the integration of combined interaction dictionaries allows for the simultaneous handling of multiple static tracks, periods without observations, and isolated track interactions. The DBN model during testing for seperate tracks predictions is shown in Figure 6.  
The use of an MJPF in the online phase is motivated by the hybrid nature of the localization task, which involves both discrete transitions across interaction clusters and continuous ego motion estimation. Traditional Kalman filters, while efficient, rely on linear motion assumptions and cannot adapt to abrupt mode transitions or temporary observation loss. On the other hand, standard particle filters, although capable of handling nonlinearity, do not explicitly model structured mode switching [51]. MJPF addresses these limitations by combining discrete state inference with continuous filtering. It uses a particle filtering mechanism to sample over a joint continuous-discrete state space and applies Kalman updates within each cluster mode.