## 3 Data Processing and Kinematic Analysis
### Object Tracking and Management
#### 3.4.3 Tracking Management and Data Association  
Data association is the step of labeling each detected object with a unique ID and associating the tracking information obtained from a LiDAR sensor that includes the estimation/prediction of the detected object's position/velocity, with each label. The data association in the tracking filter was dealt with through JPDA [23], which follows the procedure of initialization, confirmation, correction, prediction, and deletion of tracks. Among various data association algorithms, JPDA enables the calculation of all possible target association probability of each observation (see Equations (14)-(17)). The 'tentative' label was given to a newly detected track through the data association process; it was labeled as 'confirmed' if a specific track was sufficiently detected to record; the identical ID was assigned if the same track as the current one was detected. The logic flow [24] of JPDA starts with the division of detections from multiple sensors into multiple groups (however, a single LiDAR was used in this study). For each sensor, the distances from detections to existing tracks and a validation matrix were calculated by turns. After the calculation of the validation matrix, tracks and detections were separated into clusters. The following steps outline how to update each cluster:  
* S1: Generation of feasible joint events. * S2: Calculation of the posteriori probability of each joint event. * S3: Calculation of the marginal probability of each detection-track pair. * S4: Reporting of weak detections that are within the validation gate of at least one track. * S5: Unassigned and weak detections get new tracks. * S6: Tracks are deleted based on the defined number of scans without detection. * S7: All tracks are predicted to the latest time value. To formulate JPDA, we assume to have \(x_{t}^{1},\ldots,\ x_{t}^{N}\) and \(z_{t}^{1},\ldots,\ z_{t}^{M}\) as the states of \(N\) objects and \(M\) measurements at time \(t\), respectively. Here, the state vector \(x_{t}^{j}\) means the position and velocity for the \(j^{th}\) detected object and the measurements are the cluttered and noisy detected positions observed by a sensor. The probability of the data association, \(p_{t}\!\left(d_{i}^{j}\right)\) which represents the measurement index \(i\in[M]_{0}\triangleq\{0,\,1,\,\ldots,\,M\}\), is defined as follows [24]. \[p_{t}\!\left(d_{i}^{j}\right)\propto\left\{ \begin{array}{cc}(1-p_{D})\beta& \text{if }i=0\\ p_{D}\!\cdot\!N\!\left(z_{t}^{j};\hat{x}_{t}^{j},\Sigma_{S}\right)\text{ otherwise.}\end{array} \right. \tag{14}\]  
where \(p_{D}\) denotes the detection probability, \(\beta\) represents the density of the false detection, \(\hat{x}_{t}^{j}\) is the prediction state of the object, \(\sum_{S}\) is the innovation covariance matrix of the Kalman filter, and \(N\) is the normal distribution. \(p_{t}\!\left(d_{i}^{j}\right)\) is assumed to be a linear Gaussian model and its 0 value means missed or dummy detection. The joint data association space, \(\Theta\), defined in Equation (15) comprises all possible combinations of measurement-to-object assignments in order that each measurement is assigned to at most one object (A1) and each object is uniquely assigned to a measurement (A2). \[ \begin{split}\Theta&=\left\{\theta=\left(d_{i}^{i} \right)_{i\in[M]_{0},\ j\in[N]}\ \Big{|}\ d_{i}^{j}\in\{0,1\}\right.\\ &\quad\wedge\ \sum\limits_{j=1}^{N}d_{i}^{j}\leq 1,\ \forall_{i}\in[M]\text{ for }A1\\ &\quad\wedge\ \sum\limits_{j=0}^{N}d_{i}^{j}=1,\ \forall_{j}\in[N]\text{ for }A2\}.\end{split}  \tag{15}\]  
where \(|\Theta|=:n_{h}\) is the total number of joint assignments and \(\theta\) represents the binary vector that is one possible solution to the data association problem. \(\theta\in\Theta\subseteq\text{B}^{N\times(M+1)}\). Then, the marginalized JPDA probability, \(q_{t}\left(d_{i}^{j}\right)\) over \(\Theta\) can be calculated as  
\[q_{t}\left(d_{i}^{j}\right)=\sum\limits_{\theta\in\Theta_{i}^{j}}p(\theta). \tag{16}\]  
where  
\[p(\theta)=\prod\limits_{