# Traffic Demand Estimations Considering Route Trajectory Reconstruction in Congested Networks
## 2 Observed Data Analysis and Processing
### Route Trajectory Reconstruction Based on Gaussian Mixture Clustering Analysis  
Usually, only origin and destination information of a travel route can be obtained during the monitoring process of actual traffic data. For example, automatic vehicle identification data and mobile phone bill data can only capture trip start and end time and location information. Therefore, route trajectory reconstruction of such data is required for traffic demand estimation.  
The Gaussian mixture model is a soft data clustering method, which can effectively describe the distribution of the mixed density function of observed data [20]. In the process of route trajectory reconstruction, observed travel route information with the same origin and destination are subject to a set of probability distributions with unknown mean and variance. Each observation (trajectory unknown) is a sample of multi-model distribution. Next, the probability density function of the Gaussian mixture model is formed using the probability density function corresponding to multiple sets of origin and destination points. Equations (2)-(4) build a route trajectory reconstruction model based on Gaussian mixture clustering analysis:  
\[\text{max}L(c,\sigma,\pi)=\sum\limits_{i\in I}\log(\sum\limits_{k\in K^{ \omega}}\pi_{k}^{\omega}N(\hat{x_{i}}\Bigg{|}\hat{x_{k}},\sigma_{k}^{\omega 2})) \tag{2}\]  
Constraints are:  
\[\sum\limits_{k\in K^{\omega}}\pi_{k}^{\omega}=1,\forall\omega\in W \tag{3}\]  
\[0\leq\pi_{k}^{\omega}\leq 1,\forall k\in K^{\omega},\omega\in W \tag{4}\]  
where \(I\) represents the set of observed routes (trajectory unknown) \(i\in I\); \(X_{i}\) represents the \(i\)th observed route travel time; \(W\) is the set of OD pairs of the road network, \(\omega\in W\); \(K^{\omega}\) represents the set of all routes in OD pairs \(\omega,k\in K^{\omega}\); \(\pi_{k}^{\omega}\) is the mixing weight of the route \(k\) in OD pair \(\omega\), and \(\sum\limits_{k\in K^{\omega}}\pi_{k}^{\omega}=1,\forall\omega\in W\); \(c_{k}^{\omega}\) represents the expected value of travel time  
Figure 1: Performance function of link travel time. (**a**) Traditional assumption. (**b**) Actual situation.  
of the route \(k\) in OD pair \(\omega\); \(\sigma_{k}^{\omega 2}\) represents the variance in travel time of the route \(k\) in OD pair \(\omega\); \(N(\hat{x_{i}}\Big{|}c_{k}^{\omega},\sigma_{k}^{\omega 2})\) represents the probability density function of Gaussian distribution with mean \(c_{k}^{\omega}\) and variance \(\sigma_{k}^{\omega 2}\) at \(\hat{X_{i}}\). Objective function (2) represents the maximized Gaussian mixture likelihood function. Constraints (3)-(4) are mixed weight conservation and interval constraint, respectively.  
Thus, route travel time after trajectory reconstruction can be obtained using:  
\[x_{i}=\sum\limits_{k\in\mathbb{K}^{\omega}}\gamma_{k,i}^{\omega}c_{k}^{\omega},\forall i\in I \tag{5}\]  
In the following formula, \(\gamma_{k,i}^{\omega}\) represents the weight coefficient of the coincidence between the \(i\)th observed travel trajectory and the route \(k\) in OD pair \(\omega\).  
\[\gamma_{k,i}^{\omega}=\frac{\pi_{k}^{\omega}\mathcal{N}\bigg{(}\hat{x_{i}} \Big{|}c_{k}^{\omega},\sigma_{k}^{\omega 2}\bigg{)}}{\sum\limits_{k\in\mathbb{K}^{ \omega}}\pi_{k}^{\omega}\mathcal{N}\bigg{(}\hat{x_{i}}\Big{|}c_{k}^{\omega}, \sigma_{k}^{\omega 2}\bigg{)}},\forall k\in K^{\omega},\omega\in W,i\in I \tag{6}\]  
A Bayesian Traffic Demand Estimation Model Using Route Trajectory Reconstruction in Congested Networks