## 2 Methodology
### Likelihood and Model Estimation Method  
Estimation of mixture models can be classified into two methods, such as Expectation-Maximization (EM) and Bayesian methods. The EM algorithm has been widely used to estimate the mixture shape of distribution based on the maximum likelihood method. However, this method may lead to a local maximum and many starting points are needed to find a global maximum. Furthermore, since the maximum likelihood method is based on asymptotic theory, the sample size must be large [34; 35]. This means that estimating a mixture model is difficult under low sampling rates of probes using the EM algorithm. In this study, a Bayesian mixture model is used as it provides richer inferences than the maximum likelihood method; also, this approach can address parametric uncertainty and travel time properties in estimating travel time. This study employs a mixture of two Gaussian distributions since mixture models are a flexible family of models and have been used to model large heterogeneous populations. The two mixtures of Gaussian distribution with individual travel data, \(y=\{y_{1},\,y_{2}\cdots y_{n}\}\) can be expressed as follows:  
\[\mathrm{P}(y_{i}|\theta)=\sum_{k=1}^{2}w_{k}\;\mathrm{N}\Big{(}y_{i}|u_{k},\, \,\sigma_{k}^{2}\Big{)} \tag{2}\]  
where \(\;\mathrm{K}\) is the number of components and \(w_{K}\) is the proportion of component \(K\) (\(0\leq w_{K}\leq 1\), \(\sum_{k=1}^{2}w_{K}=1\)). \(\mathrm{N}(y_{i}|u_{k},\,\sigma_{k}^{2})\) is the normal distribution with mean \(\mu_{k}\) and variance \(\sigma_{k}^{2}\). \(\theta\) is the vector of all parameters, \(\theta=\{(w_{1},w_{2}),(\mu_{1},u_{2}),(\sigma_{1}^{2},\sigma_{2}^{2})\}\).  
The likelihood function of the mixture distribution with latent variable \(Z\) for classification can be expressed as follows. Latent variables are variables that are not directly observed but inferred from other observed variables. Thus, the latent variable \(Z\) follows the multinomial distribution and the likelihood function is expressed with the latent variable.  
\[(y,\,Z|\theta)=\prod_{i=1}^{n}\mathrm{P}\big{(}y_{i}|Z_{i}\theta\big{)}\mathrm{ P}(Z_{i}|\theta)=\prod_{i=1}^{n}\sum_{k=1}^{K}[w_{k}\mathrm{P}\Big{(}y_{i}|\mu_{k}, \,\sigma_{k}^{2}\Big{)}]^{Z_{i,\;k}} \tag{3}\]  
where \(Z_{i,\;k}\) is the ith element of the kth component and, if \(y_{i}\) follows \(\mathrm{N}\) (\(\mu_{k},\,\sigma_{k}^{2}\)), \(Z_{i,\;k}\) is 1; if not, \(Z_{i,\;k}\) is 0.  
The Bayesian mixture model is applied when the number of components is assumed to be known. According to the influence of traffic oscillation, this paper used the Bayesian mixture model with two components to estimate the travel time distribution. To classify the individual travel data of the kth component, the conditional probability function can be expressed as follows:  
\[\mathrm{P}\Big{(}Z_{i}=k|w^{(r-1)},\mu^{(r-1)},\sigma^{2(r-1)},c\Big{)} \propto\;w^{(r-1)}.\mathrm{P}\Big{(}y_{i}\mid\mu^{(r-1)},\sigma^{2(r-1)} \Big{)}, \tag{4}\]  
where \(w^{(r-1)},\mu^{(r-1)},\sigma^{2(r-1)}\) are the parameters of the Gaussian mixture model at \(r-1\) iterations and \(y_{i}\) is the individual travel data for classification.