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Interpretation of common-yet-challenging interaction scenarios can benefit well-founded decisions for autonomous vehicles. Previous research achieved this using their prior knowledge of specific scenarios with predefined models, limiting their adaptive capabilities. This paper describes a Bayesian nonparametric approach that leverages continuous (i.e., Gaussian processes) and discrete (i.e., Dirichlet processes) stochastic processes to reveal underlying interaction patterns of the ego vehicle with other nearby vehicles. Our model relaxes dependency on the number of surrounding vehicles by developing an acceleration-sensitive velocity field based on Gaussian processes. The experiment results demonstrate that the velocity field can represent the spatial interactions between the ego vehicle and its surroundings. Then, a discrete Bayesian nonparametric model, integrating Dirichlet processes and hidden Markov models, is developed to learn the interaction patterns over the temporal space by segmenting and clustering the sequential interaction data into interpretable granular patterns automatically. We then evaluate our approach in the highway lane-change scenarios using the highD dataset collected from real-world settings. Results demonstrate that our proposed Bayesian nonparametric approach provides an insight into the complicated lane-change interactions of the ego vehicle with multiple surrounding traffic participants based on the interpretable interaction patterns and their transition properties in temporal relationships. Our proposed approach sheds light on efficiently analyzing other kinds of multi-agent interactions, such as vehicle-pedestrian interactions. View the demos via https://youtu.be/z_vf9UHtdAM.