# Traffic Demand Estimations Considering Route Trajectory Reconstruction in Congested Networks
## 2 Observed Data Analysis and Processing
### Analysis of the Relationship between Traffic Demand and Observed Variables  
Travelers' choice of travel destination and driving route can be regarded as a large number of independent Bernoulli experiments. Therefore, this paper assumes that the traffic demand follows the multivariate normal distribution, \(\mathbf{D}\sim\text{MVN}(\mathbf{\mu_{D}},\mathbf{\Sigma_{D}})\), with mean vector \(\mathbf{\mu_{D}}\) and covariance matrix \(\mathbf{\Sigma_{D}}\). Traffic demand varies in different time periods and weather conditions. For example, traffic demand increases during holidays and decreases in severe weather conditions. In order to describe these characteristics, this paper adopts the following form of traffic demand:  
\[D^{\omega}=k^{\omega}U+\eta^{\omega} \tag{7}\]  
where \(k^{\omega}\) describes the relative weight that the demand of OD pairs \(\omega\) account for total traffic demand, and \(\mathbf{k}\) is its vector form, namely \(\mathbf{k}=\{\cdots,k^{\omega},\cdots\}\). \(U\) reflects the changing situation of traffic demand over time and weather, and follows the normal distribution form of mean \(\mu_{U}\) and variance \(\sigma_{U}^{2}\). \(\eta^{\omega}\) is random errors which are mutually independent, and follows the normal distribution form of mean 0 and variance \(\sigma_{\eta}^{2}\), and \(\mathbf{\eta}\) is its vector form, namely \(\mathbf{\eta}=\{\cdots,\eta^{\omega},\cdots\}\).  
Considering the conservation relationship between link flow and OD demand:  
\[v_{a}=\sum\limits_{\omega\in W}\sum\limits_{k\in\mathbb{K}^{\omega}}p_{k}^{ \omega}\delta_{a,k}^{\omega}d^{\omega},\forall a\in A \tag{8}\]  
where \(v_{a}\) represents traffic flow of link \(a\) and \(\delta_{a,k}^{\omega}\) represents the associated parameter of route-link among OD pairs \(\omega\), if link \(a\) is on route \(k\), the parameter is 1; otherwise, the parameter is 0. \(p_{k}^{\omega}\) represents travelers' route choice probability on route \(k\) in OD pair \(\omega\), which can be obtained by solving the following stochastic user equilibrium (SUE) model:  
\[p_{k}^{\omega}=\frac{\exp\big{(}-\theta c_{k}^{\omega}\big{)}}{\sum\
olimits_ {k}\exp\big{(}-\theta c_{k}^{\omega}\big{)}},\forall k\in K^{\omega},\omega\in W \tag{9}\]  
where \(\theta\) is a discrete parameter used to measure the perception error degree of travelers; and \(c_{k}^{\omega}\) represents the estimated route travel time on route \(k\) in OD pair \(\omega\).  
\(\beta_{a}^{\omega}=\sum_{k}p_{k}^{\omega}\delta_{a,k}^{\omega}\) represents the ratio between the number of drivers using link \(a\) and the demand of OD pairs, where \(\beta\) is all OD-link ratio matrix. Thus, Equation (8) can be written as:  
\[v_{a}=\sum_{\omega\in W}\beta_{a}^{\omega}d^{\omega},\forall a\in A \tag{10}\]  
To reduce the solving complexity of the Bayesian estimation model, the first-order Taylor expansion of road impedance function replaces link travel time. As the designed algorithm is iterative, the error generated by the replacement in the iterative algorithm gradually decreases. According to the link congestion and impedance function (Equation (1)), link travel time can be deduced as:  
\[t_{a}=m_{a}v_{a}+n_{a}=m_{a}\sum_{\omega\in W}\sum_{k\in\mathbb{K}^{\omega}}p_{ k}^{\omega}\delta_{a,k}^{\omega}d^{\omega}+n_{a},\forall a\in A \tag{11}\]  
where \(m_{a}\) and \(n_{a}\) are the coefficients of first-order Taylor expansion. Further, route travel time can be written as:  
\[c_{k}^{\omega}=\sum_{a\in A}\delta_{a,k}^{\omega}t_{a},\forall k\in K^{\omega },\omega\in W \tag{12}\]