We propose new measures for assessing traffic conflicts, called “Potential Response Inspired Conflict (PRIC)”, taking the counterfactual sentence “there is a risk of collision if their movements remain unchanged” into account, based on the potential outcome models. Such a counterfactual sentence is used in the widely acceptable definition of traffic conflict given by ICSTCT (Amundsen and Hyden, 1977). First, we point out that most of existing traffic risk evaluation measures may not take such a counterfactual sentence into account.To solve this problem, we introduce the potential outcome model into traffic conflict technique, in order to formulate the PRICs. In addition, we provide three identification conditions for the PRICs, and show that the existing traffic risk evaluation measures can statistically reflect the definition of the traffic conflict through the PRICs under certain conditions. Furthermore, when the proposed identification conditions do not hold, we formulate the bounds on the PRICs under certain causal assumptions.Finally, through the application of the PRICs to numerical examples and “The 100-Car Naturalistic Driving Study (Dingus et al., 2006)”, we discuss the usefulness and limitation of the traffic conflict measures.

We consider the problem of clustering functional data using nonlinear mixed effects models along with the technique of basis expansions. With the help of fixed and random effects functions, the nonlinear mixed effects model makes it easy to handle unbalanced or sparse data which are highly occurred in the longitudinal study. We assume different numbers of basis functions for fixed and random effects functions. Unknown parameters included in the model are estimated by the maximum likelihood method along with the EM algorithm, and then the numbers of basis functions included in the model are selected by model selection criteria.
We then apply hierarchical and non-hierarchical clustering methods to the predicted coefficients of the random effect terms of functional data in order to highlight the features of each subject. The hierarchical clustering such as the Ward's method proceeds in successive steps from smaller to larger clusters, which can be directly observed visually. In contrast, the non-hierarchical clustering such as the self-organizing maps consists of progressively refining the data partitions to obtain a given number of clusters. In functional cluster analysis, we can remove the measurement errors of observed data and therefore we can capture the functional structure behind the data. We report the results of application of the proposed method to some real data sets such as environmental data and weather data.