## 2 Materials and Methods
### Method
#### 2.2.2 Introduction of Information Divergence  
Information divergence is an important concept in information science, commonly used to measure the difference or similarity between two probability distributions.  
In information theory, probability distributions are typically represented as either a vector or a matrix, with each dimension of the vector or each element of the matrix representing the probability of an event. The definition of information divergence leverages the concepts of information entropy and KL divergence (Kullback-Leibler divergence). KL divergence, also known as relative entropy, is a method for measuring the difference between two probability distributions, with a smaller value indicating greater similarity between the distributions. Information divergence is defined as the relative entropy between two probability distributions minus the difference in their entropies. Here, relative entropy is utilized to quantify the difference between the two probability distributions, while entropy is used as a measure of uncertainty. Information divergence can be expressed as Equation (1):  
\[D(A||B)=\sum\bigl{(}A_{ij}log\bigl{(}A_{ij}/B_{ij}\bigr{)}-A_{ij}+B_{ij} \tag{1}\]  
where \(A\) and \(B\) are two matrices of the same dimensions. For the problem of spatial-temporal co-clustering, we can set the original matrix as \(A\) and hope to find an approximate clustering matrix \(B\) that minimizes the information divergence between \(A\) and \(B\). This process can be carried out through iterative optimization.  
In spatial-temporal co-clustering algorithms, employing information divergence as the optimization function is an attractive choice, particularly because the input for spatial-temporal co-clustering is a matrix. First, a significant advantage of using information divergence is its adaptability; it does not depend on the specific distribution form of the data, allowing the algorithm to flexibly adapt to various types of data. This adaptability enables spatial-temporal co-clustering algorithms based on information divergence to be widely applied to text data, spatial-temporal data, and other complex data types, enhancing the algorithm's universality. Secondly, information divergence preserves the informativeness of data. During the clustering process, information divergence optimizes clustering results by minimizing differences within clusters and maximizing differences between clusters, ensuring that data post-clustering retains as much information from the original data as possible. This method is particularly valuable in the field of pattern recognition. Lastly, strong interpretability is also a key advantage of information divergence. By quantifying the difference between two probability distributions, information divergence provides an intuitive way to understand clustering results.  
In summary, using information divergence as the optimization function in spatial-temporal co-clustering algorithms offers powerful adaptability, preserves the informativeness of data, provides good interpretability, and captures spatiotemporal distribution characteristics. These advantages make it an effective tool for addressing complex data clustering challenges.