## 2 Algorithmic Basis for Urban Residential Vacancy Rate Observation
### Application of the Algorithm in Urban Residential Vacancy Rate Observation
By constructing a deep neural network, the algorithm can automatically learn and extract complex features, enhancing the generalization ability of the observation model [75; 76; 70]. In particular, in the application of high-resolution remote sensing images, deep learning algorithms such as the image generation technology of generative adversarial networks (GANs) can generate realistic residential area images when data are insufficient, assisting in the judgment of vacancy conditions [77; 78]. The application of algorithms in urban residential vacancy rate observation has achieved remarkable results. The application of different algorithms in urban residential vacancy rate observation follows a well-thought-out logic. Machine learning algorithms, with their strong nonlinear modeling capabilities, are well suited to capture the complex relationships between various factors and the vacancy rate. For example, decision trees, support vector machines and random forests can analyze data from multiple dimensions, such as real estate transaction records and census data. By doing so, they can identify key factors influencing the vacancy rate, like the impact of population structure on the vacancy rate of different-sized apartments. This ability to handle complex relationships is crucial because the factors affecting urban residential vacancy rates are highly non-linear. Multi-source data fusion algorithms play a vital role in integrating diverse data sources. Data from different origins, such as drone aerial images, census data and real estate transaction records, each carry unique information. By fusing these data sources through preprocessing, feature extraction and similarity measurement, we can obtain a more comprehensive and accurate picture of the vacancy rate. This not only improves the accuracy of vacancy rate observation but also reveals the regional distribution of residential vacancy, providing a scientific basis for policy making. Adaptive algorithms are designed to address the dynamic nature of urban development. In the process of urban new district development, the residential vacancy rate is influenced by various factors like planning, transportation and supporting facilities. Adaptive algorithms can adjust the observation model in real time according to these changing factors, ensuring the timeliness and accuracy of the prediction. Deep learning algorithms, with their ability to automatically learn and extract complex features, enhance the generalization ability of the observation model. Inthe application of high-resolution remote sensing images, deep learning algorithms like generative adversarial networks (GANs) can generate realistic residential area images when data are insufficient, assisting in the judgment of vacancy conditions. Collectively, these algorithms form a comprehensive approach to address the challenges in accurately observing urban residential vacancy rates. This integrated approach is essential for urban planning, real estate management and policy making, as it provides more accurate and reliable data support, enabling more informed decision-making processes. The combination of machine learning, multi-source data fusion, adaptive and deep learning algorithms not only improves the accuracy and efficiency of observation, but also provides strong technical support for coping with complex environments, processing diverse data sources and meeting urban development needs [79]. In practical applications, the continuous optimization and improvement of these algorithm models are of great significance in improving generalization ability, reducing computational complexity and enhancing model interpretability.