## 3 Algorithmic Challenges in Complex Environments  
### Data Characteristics in Complex Environments  
In the urban environment, algorithms face challenges from complex environments when observing residential vacancy rates. The data have significant spatiotemporal heterogeneity, with obvious differences in different areas. It is affected by multiple factors like macro-economy, policy, market and residents' income, making it highly complex. Data sources are diverse, with issues like inconsistent coverage, accuracy and timeliness, and may contain missing or incorrect data. Also, residents' behavior is unpredictable, adding to the complexity. Therefore, algorithms need strong adaptability and robustness to handle these challenges [80; 81; 82]. As shown in Table 2, these data features are derived from the imbalance of urban development, the diversity of regional functions and the variability of the residential market, which together constitute the challenges that the observation algorithm needs to deal with. The data characteristics in a complex environment are manifested in the following aspects: First, the data characteristics of urban residential vacancy rates show significant temporal and spatial heterogeneity. There are obvious differences in residential vacancy conditions in different cities, regions and even streets, which requires the algorithm to effectively identify and adapt to the differences in local characteristics [83; 84]. For example, some newly built areas may have a high residential vacancy rate because supporting facilities are not yet complete, while some mature communities may have a relatively low vacancy rate because of convenient living. The algorithm needs to take these spatiotemporal factors into account and reasonably assign weights and extract features for data in different regions. Secondly, the urban residential vacancy rate is affected by many factors, including macroeconomics, policy regulation, market supply and demand, and residents' income level, which make the data characteristics highly complex. In algorithm design, how to select key variables from many influencing factors and accurately capture the relationship between them and the vacancy rate has become a major challenge. Inaddition, there may be interactions between these factors, which further increases the difficulty of analyzing data features. Furthermore, the multi-source nature of data sources brings new challenges to the algorithm. The data sources for observing residential vacancy rates include but are not limited to real estate market transaction data, census data, remote sensing images and urban infrastructure information [85; 86; 87; 88]. These data sources vary in coverage, accuracy, timeliness, etc., and may even have problems such as missing data and errors. To address these problems, the algorithm needs to have strong robustness and data fusion capabilities, be able to extract effective information from multi-source data, and reduce the uncertainty brought by a single data source. In addition, the dynamic nature of urban development also places higher demands on algorithms. With the advancement of urbanization, residential vacancy rates will be affected by urban planning, transportation development, industrial layout and other factors, showing a dynamic trend. The algorithm needs to have a certain predictive ability and be able to reasonably predict future vacancy rate changes based on historical data and existing trends to assist government departments and real estate companies in making effective decisions. Numerous studies have shown that there are significant nonlinear components in the field of urban residential vacancy rate observation. Take the research of a first-tier city as an example. Researchers collected the residential vacancy rate data of this city over the past decade.