## 2 Algorithmic Basis for Urban Residential Vacancy Rate Observation
### Algorithm Overview  
In the research and application of urban residential vacancy rate observation, algorithms play a vital role. They provide key technical support for coping with the challenges of complex environments, diverse data sources and urban development needs [40]. We have organized the types of algorithms for urban residential vacancy rate observation into Figure 1.  
Traditional statistical learning methods, such as linear regression, logistic regression and time series analysis, are based on historical data mining and analysis to build residential vacancy rate prediction models [41; 42]. For example, linear regression aims to find the best-fit linear relationship between independent variables (like economic growth rate, population density) and the dependent variable (residential vacancy rate). It uses the least-squares method to minimize the sum of the squared differences between the observed and predicted values. This approach assumes a linear relationship, which is relatively simple and computationally efficient. However, in real-world scenarios, especially in urban residential vacancy rate observation, the relationship is often more complex. For instance, when considering the impact of multiple factors on the vacancy rate, linear regression may not accurately capture the nonlinear interactions between variables. For example, urban migration patterns often exhibit nonlinear relationships with vacancy rates. A sudden influx of young professionals may initially reduce vacancy rates in downtown areas but later lead to oversupply as new housing projects are rapidly developed, creating a U-shaped curve that linear regression cannot capture. Similarly, infrastructure investments (e.g., subway extensions) may have delayed effects on vacancy rates, with  
Figure 1: Flowchart of algorithms for urban residential vacancy rate observation. impacts varying by neighborhood density and socioeconomic composition. While polynomial regression can approximate certain nonlinear relationships (e.g., quadratic effects of housing prices on vacancy rates), it struggles with high-dimensional data and overfitting in complex urban systems. For example, capturing interactions between migration, policy, and infrastructure would require polynomial terms that quickly become computationally infeasible and difficult to interpret. Such methods have certain advantages in dealing with linear problems and data with clear regularity. However, with the increasing complexity of the urban environment, the nonlinear characteristics of the data are becoming increasingly prominent, making it difficult for these traditional methods to meet actual needs in terms of prediction accuracy [43; 44]. Despite their limitations in nonlinear scenarios, traditional statistical methods offer critical strengths, including interpretability and computational efficiency. For example, linear regression provides transparent insights into the relationship between variables, enabling policy makers to directly understand how factors like population density or housing prices influence vacancy rates. This clarity is invaluable for actionable decision making in urban planning [45; 46]. Additionally, these methods are less prone to overfitting when working with limited datasets, a common challenge in urban vacancy rate studies where comprehensive data may be scarce. Moreover, their low computational demands make them accessible in resource-constrained environments or for rapid prototyping. In particular, when faced with complex nonlinear problems and a large amount of noisy data, the limitations of traditional statistical learning methods become more apparent. A study by [PERSON] et al. applied linear regression to predict vacancy rates in Beijing using economic and demographic variables but found a poor fit (R\({}^{2}\) = 0.45), highlighting its inability to model interactions between housing policies and market speculation [47]. In contrast, machine learning models achieved R\({}^{2}\) > 0.78 by capturing nonlinear dynamics.