## 1 Introduction  
In recent years, China has experienced a rapid urbanization process, with significant changes in urban population density and spatial structure. By 2024, China's urbanization rate reached 67% [1], but the distribution of urban population density is extremely uneven, and the eastern region is densely populated, while the western region is sparsely populated. This uneven population distribution presents a significant challenges to urban planning and community policy [2]. In the context of rapid urbanization in China, the phenomenon of a \"ghost city\" has attracted wide attention. The so-called \"ghost city\" refers to a new city with a sparse population and an extremely high housing vacancy rate due to excessive development. Take Kangbashi New Area in Ordos as an example; it was called a \"ghost city\" due to the lack of population support at the beginning, but with government intervention and the gradual improvement of the market mechanism, it gradually became livable [3]. This phenomenon reflects the dynamic change of vacancy rates in the process of Chinese urbanization and its impact on urban sustainability. Vacancy rate is an important indicator of urban sustainable development, and there are many relevant studies in the world. For example, [PERSON] et al. proposed the vacancy rate visual analysis method (VVAM) to evaluate the effectiveness of an urban regeneration strategy by quantifying and visualizing building vacancy rates [4]. In addition, [PERSON] et al. conducted a longitudinal analysis of vacancy rates in large cities in the United States from 1960 to 2010 and found that factors such as demographic change, unemployment rate, and urban expansion had a significant influence on vacancy rates in [5]. These studies provide an important international perspective for understanding the dynamic changes in vacancy rates. The term \"vacant housing\" denotes properties that are unoccupied or are actively listed for sale or rent [6; 7]. Vacancies can be transient (e.g., seasonal or transitional) [8] or structural (e.g., long-term, driven by economic or policy factors) [9]. This issue not only results in the wastage of resources [10] and escalates social and economic costs [11] but it may also compromise public safety [12; 13]. The housing vacancy rate in a given region is a significant indicator of the health of the local real estate market. Consequently, employing scientific methodologies to investigate the housing vacancy rate can facilitate the resolution of this issue, enhance rational allocation and utilization of resources, and foster sustainable urban development. Since most developed countries develop early, they usually have a housing vacancy rate calculation system based on actual statistical data. For instance, the United States utilizes the \"American Community Survey, Current Population Survey/Housing Vacancies and Homeownership Survey (CPS/HVS) [14; 15]\", the \"American Community Survey (ACS)\" [16], and records from the \"United States Postal Service (USPS) [17]\". Japan conducts sampling surveys of houses and land to compile the Housing and Land Survey (HLS) data [18], while Singapore's Urban Redevelopment Authority (URA) compiles relevant data under its leadership [19]. In contrast, Chinese authorities have yet to establish a formal statistical system, and the difficulty of obtaining data from the real estate industry has led to relatively lagging progress in the research of the urban housing vacancy rate in China. At present, some scholars in the academic community have conducted vacancy rate studies based on survey statistics in certain regions of China. For instance, [PERSON] took Shanghai as the research object, designed a field survey plan, and carried out investigations [20]. In addition, there are studies that, based on electricity consumption data, set a minimum threshold and extract the vacancy rate [21]. However, these types of research rely on a large amount of actual survey data.