## 2 Algorithmic Basis for Urban Residential Vacancy Rate Observation
### Algorithm Overview
Machine learning methods, including support vector machines, decision trees and random forests, have shown great advantages in capturing nonlinear relationships in data due to their strong nonlinear modeling capabilities [48; 49; 50; 51]. Support vector machines (SVMs) work by mapping the input data into a high-dimensional feature space and finding an optimal hyperplane that maximally separates different classes. In the context of residential vacancy rate observation, SVMs can classify whether a house is vacant or occupied based on various features like building attributes, location and market conditions. Decision trees, on the other hand, recursively split the data based on different features. Each internal node represents a feature test, and the branches represent the outcomes of the test. For example, a decision tree might first split the data based on the building age, then further split the data based on the location, to predict the vacancy rate. Random forests are an ensemble of decision trees. They build multiple decision trees on different subsets of the data and combine their predictions, which helps to reduce overfitting and improve the overall prediction accuracy. In urban residential vacancy rate analysis, random forests can handle large-scale and complex data, identifying important factors affecting the vacancy rate and their interactions. In machine learning algorithms for predicting urban residential vacancy rates, typical input variables are crucial. They mainly include the following:  
Geospatial variables: These include the longitude and latitude of residential properties, which help the algorithm identify regional vacancy rate differences. Building attribute variables: Building age, structure type and area size matter. New and old buildings have different vacancy rates due to various factors. Different structural types affect living comfort and costs, influencing vacancy. Also, large- and small-area properties have different market demands and vacancy rates. Socioeconomic variables: Regional economic development (measured by per capita GDP (Gross Domestic Product), industrial structure), population density, residents' income level and employment situation impact the vacancy rate. Developed regions have lower vacancy rates. High-income groups' high housing-quality requirements and low-income groups' limited housing-buying/renting ability, along with employment factors, affect housing demand and vacancy. Real estate market variables: Housing price trends, rental levels and the market supply-demand relationship influence the vacancy rate. Rising prices may lead to property hoarding, while rental levels and supply-demand ratios directly affect vacancy. Deep learning algorithms, such as convolutional neural networks, recurrent neural networks and long short-term memory networks, effectively address the challenges brought by diverse data sources through automatic learning of feature representation and hierarchical learning capabilities [52; 53; 54]. These models show strong performance when processing large amounts of high-dimensional, complex data, making the observation of residential vacancy rates more accurate. However, deep learning methods also have certain limitations, such as complex training processes and high computational costs, which have brought certain problems to practical applications. Ensemble learning methods, such as bagging, boosting and stacking, have been widely used in the observation of urban residential vacancy rates in China by combining multiple prediction models to improve prediction accuracy [55; 56; 57]. This type of method can make full use of the advantages of various algorithms, improve the stability and reliability of observation results and provide strong support for the regulation of urban residential vacancy rates. Driven by big data and cloud computing technologies, data-driven algorithms have made important progress and achieved accurate prediction of residential vacancy rates.