## 1 Introduction  
With the rapid development of the social economy, intercity connections have become increasingly close. Building metropolitan areas has emerged as a global strategy for regional development, especially in China where dozens of metropolitan areas have been built in the past two decades. Transportation infrastructure, particularly railways, plays a pivotal role in this process. Based on the function, railways are divided into four types [1], as shown in Table 1, including the trunk railway serving long-distance travels, the intercity railway serving metropolitan travels, the intracity railway serving travels between the urban areas and suburbs of cities, and the urban rail transit (URT) serving travels in the urban areas of cities. Traditionally, intercity travel relied on trunk railways and intercity railways. However, a new trend has emerged: URT systems now extend across city boundaries. For example, China's first intercity metro (Guangfo Metro) opened in 2010, followed by lines like Shanghai Metro Line 11 and Wuhan Metro Line 11. Compared to traditional railways, URTs have significant advantages [2] in terms of departure frequency, departure intervals, ticket, etc., which have effectively promoted connections between cities [3]. With the increasing demand for intercity travel, many scholars have studied the characteristics of intercity ridership [4; 5]. Regarding the railway network, [PERSON] et al. [6] analyzed railway ticketing data in the Guangdong-Hong Kong-Macao Greater Bay Area, revealing a steady rise in intercity ridership from 2015 to 2019. The growing demand for station connectivity and the operation of the Guangzhou-Shenzhen-Hong Kong high-speed railway significantly strengthened ties between Guangzhou and Shenzhen. Aiming at the intercity URT, [PERSON] et al. [7] conducted a questionnaire of the Guangfo intercity metro and analyzed its characteristics. The results showed that the intercity ridership had an unbalanced distribution in space. Clustering is often adopted to identify travel patterns. And the clustering analysis in public transit based on the smart card data often consider two standpoints: a station-oriented point of view and a ridership-focused operational one [8]. From the station-oriented point of view, [PERSON] et al. [9] used the smart card data of the metro in Nanjing, China, and proposed a similarity-based k-medoids clustering approach to reveal the ridership patterns. Then, the correlation analysis between the clustering results of metro stations and surrounding land uses was used to identify land use surrounding the stations. From the ridership-focused point of view [10; 11], [PERSON] et al. [12] proposed a ridership forecasting method based on metro ridership clustering as well, which used the hierarchical clustering algorithm, and proved its effectiveness. Considering the clustering method, K-means and hierarchical clustering algorithms have been widely used in clustering, but it is difficult to cluster data with special structures [13]. In this regard, a feasible way to cluster data distribution is to adopt a finite mixture model. The Gaussian mixture model (GMM) [14] is a widely used clustering method. It divides data into several groups with different Gaussian distributions [15; 16]. And many studies have demonstrated that the GMM outperforms some traditional clustering methods [17; 18]. While traditional travel survey data have significantly contributed to understanding travel patterns, they have notable limitations such as being prone to errors, lacking cost-efficiency, and lacking detailed intermediate trip information [19]. In the age of big data, all kinds of innovative data have been applied to supplement or even replace traditional survey data in travel patterns research, which mainly include GPS data, smart card data [20; 21], and mobile phone data. Among them, the smart data in transportation are widely deployed for automatic fare collection systems in public transits [22], such as bus [23] and metro [24]. [PERSON] et al.