## 3 Methodology  
### Latent Profile Analysis  
In the selection of the classification method for this study, latent profile analysis (LPA) was chosen for four principal reasons. Firstly, LPA, as an individual-centered analytical technique utilizing data and probability models, effectively discerns latent model classifications among individuals [68]. This method has demonstrated success across various fields [69], offering a viable solution for driver classification in our study. Secondly, the adoption of LPA is underpinned by its adherence to specific-fit index standards, providing more accurate outcomes compared to traditional classification approaches. Furthermore, the inherent flexibility of latent class clustering eliminates the need to pre-specify the number of groups or label individual observations, allowing for the determination of the optimal number of clusters through diverse statistical criteria. Lastly, LPA's capability to accommodate various types of variables, such as counts, continuous, categorical, and nominal variables, without the necessity for standardization [70] renders it a judicious selection within the context of this research.  
LPA has been increasingly used as an analysis approach over the past few years, and it is becoming more common in applied research, especially in marketing. In the same vein, it has been widely used in the transportation field.  
The overall goal of LPA is to uncover latent profiles or groups (\(k\)) of individuals (\(i\)) who share a meaningful and interpretable pattern of responses on their measures of  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
interests (_j_). This is performed using joint and marginal probabilities in within-cluster and between-cluster models.  
The within-class models are expressed as follows:  
\[y_{ij}=\mu_{j}^{(k)}+\epsilon_{ij} \tag{1}\]  
\[\epsilon_{ij}\sim N\left(0,\sigma_{j}^{2(k)}\right) \tag{2}\]  
where \(\mu_{j}^{(k)}\) denotes the model implied mean; \(\sigma_{j}^{2(k)}\) represents the model implied variance, which will vary across \(j\) = 1, 2,..., J outcomes and \(k\) = 1, 2,..., \(K\) classes or profiles. The general assumptions of LPA include that of outcome variables being normally distributed within each class and the above-mentioned within-class outcomes being locally independent.  
The between-class model represents the probability of membership in a given class \(k\), and it is defined as follows:  
\[p(c_{i}=k)=\exp\left(\omega^{(k)}/\sum_{k=1}^{K}\exp\left(\omega^{(k)}\right)\right) \tag{3}\]  
where \(\omega^{(k)}\) denotes a multinomial intercept (fixed at 0 for the final class); \(c_{i}\) represents the latent classification variable in terms of the individual. On that basis, the within-class and between-class models can be incorporated into a single model based on the law of total probability, such that it yields  
\[f(y_{i})=\sum_{k=1}^{K}p(c_{i}=k)f(y_{i}|c_{i}=k) \tag{4}\]  
The above equation expresses the marginal probability density function for an individual (_i_) after summing across the joint within-class and between-class density probabilities for the J outcome variables, weighted by the probability of class or profile membership from Equation (3). Lastly, the LPA analysis results in a posterior probability for the respective individual are defined in Equation (5):  
\[t_{ik}=p(c_{i}=k|y_{i})=\frac{p(c_{i}=k)f(y_{i}|c_{i}=k)}{fy_{i}} \tag{5}\]  
The above equation represents the probability of an individual (_i_) being assigned membership (\(c_{i}\)) to a specific class or profile (_k_), given their scores on the outcome variables in the \(y_{i}\) vector. A posterior probability (_t_) was determined for the respective individual in the respective profile. To be specific, values closer to 1.0 indicate a higher probability of membership to a specific profile. With an increase in difference between the posterior probabilities for an individual, the certainty regarding their membership assignment turns out to be increased.