# A Cox Model-Based Workflow for Increased Accuracy in Activity-Travel Patterns Generation
## 2 Literature Review
### Modeling Activity Durations  
In the context of estimating activity durations, literature predominantly emphasizes hazard-based models, which are among the most widely adopted approaches in activity-based travel behavior research. These models estimate the hazard function, that is, the instantaneous probability that an activity will end at a given time, conditional on it having lasted until that point. Early studies primarily employed parametric hazard-based models, which assume a specific probability distribution for the duration process, most commonly the Weibull or Exponential distributions. These models offer analytical tractability and interpretability but are limited by their reliance on predefined functional forms, which may not adequately capture complex duration patterns observed in empirical data. Some key studies in this area include Sreela [34], which was designed specifically for shopping durations and focuses on the influence of various variables. Additionally, [PERSON] [35] and [PERSON] and [PERSON] [36] primarily study shopping durations, while also considering other activity purposes. Their work investigates whether unobserved factors not captured in the available data influence results and how such factors should be appropriately accounted for. Another parametric hazard-based model is that of [PERSON] and [PERSON] [37], which utilizes GPS data to correlate activity purpose with duration. Beyond parametric models, semi-parametric hazard-based models were also explored. The key difference between these and parametric models is that semi-parametric models do not assume a specific probability distribution for the hazard function. Instead, the baseline hazard function remains unspecified and is estimated from the data (e.g., individual characteristics). One such model is that of [PERSON] et al. [38], which addresses a gap in the literature by analyzing activity durations during emergency situations (such as COVID-19). This study incorporates socioeconomic and demographic characteristics, as well as land-use variables, as major influences on activity durations. Another semi-parametric model is the Cox model developed by [PERSON] and [PERSON] [5], which is based on a five-year study and provides new insights into the dynamic nature of time in activity durations. However, none of these studies explicitly examined the effect of unobserved heterogeneity. Another hazard-based model studied was that of [PERSON] et al. [39]. This research employs an accelerated hazard model to estimate the relationships between social activity durations and the interpersonal relationships of the participants. It also examines details such as the frequency of these activities and the conditions under which they were planned. The main distinction of this model is that variables directly influence activity duration, unlike previous models, where variables primarily affected risk factors, which in turn influenced duration. In addition to hazard-based models, a discrete-continuous choice model by [PERSON] et al. [40] was also studied. This model simultaneously represents two levels of choice: a discrete choice regarding participation in a maintenance activity and a continuous variable corresponding to the duration of the activity. This approach highlights the interdependence between these two aspects of the problem as modeled by the two variables. Finally, another alternative approach to modeling activity durations was examined in the work of [PERSON] and [PERSON] [41]. They used a path model to emphasize causal relationships between variables, aiming to capture both direct and indirect effects on activity durations. To summarize the literature and underline the contribution of our study, Table 1 is presented below. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  
Our study is closely connected to the work of [PERSON] [5], [PERSON] [38], and [PERSON] [30] in terms of activity duration modeling.