## 3 Selection of Input Variables for Influencing Factors of Mechanical Properties of UHPC  
### Analysis of the Influence of Single Factors on the Compressive Strength of UHPC  
This section employs the heatmap method in Python to analyze the correlation between various input variables and the compressive strength of concrete, as illustrated in Figure 2. This technique visually represents the degree of correlation between the variables using an intuitive heatmap. To enhance the necessary details for applying the heatmap method to other material science fields, a Python program and a detailed explanation for creating a heatmap are shown in Appendix A. We offer the steps to visualize the correlation matrix for different input variables and their relationships, which can be used for predicting material properties such as compressive strength. Detailed explanation is as following:  
1. Data Preparation  
The dataset utilized in this analysis is stored within a pandas DataFrame, where each column represents a distinct variable, including both input features and the target variable. The data is typically loaded from a CSV file or another comparable data source into the pandas DataFrame, facilitating efficient manipulation and analysis. 2. Correlation Matrix  
To quantify the relationships between variables, the Pearson correlation coefficients are computed using the 'data.corr()' function. This function generates a correlation matrix that elucidates the degree of linear relationship between each pair of variables, thereby enabling a comprehensive understanding of the interdependencies within the dataset. 3. Heatmap Creation  
The visualization of the correlation matrix is achieved using the 'heatmap' function from the seaborn library. This function generates a heatmap, where the correlation coefficients are directly annotated onto the heatmap cells by setting the 'annot = True' parameter. The heatmap employs a 'coolwarm' color map to intuitively differentiate between positive and negative correlations, with warmer colors typically representing positive correlations and cooler colors indicating negative correlations. Figure 2: Heatmap of variable correlations. (4) Customization  
To ensure that the heatmap is both interpretable and comparable across different studies, several customization parameters are employed. The 'vmin', 'vmax', and 'center' parameters standardize the color intensity across the dataset, ensuring a consistent visual scale. Additionally, the 'plt.title', 'plt.xlabel', and 'plt.ylabel' functions are used to add descriptive labels and titles to the heatmap, enhancing its clarity and utility in communicating the results. (5) Display  
The final step in the process is the display of the heatmap, which is accomplished using the 'plt.show()' function. This function renders the heatmap, making the correlation matrix accessible for interpretation and analysis. The heatmap visually represents the correlation matrix, where each cell's color intensity signifies the strength and direction of the correlation between two variables. A darker shade indicates a stronger correlation. This graphical representation aids in identifying key predictors and understanding their relationships, thus enhancing the predictive modeling process for UHPC compressive strength. Through detailed heatmap analysis, seven key variables were identified as having the most significant impact on the predictive model for concrete compressive strength: water content, cement content, steel fiber content, superplasticizer content, fly ash content, silica fume content, and slag powder content. This identification allows for the optimization of the predictive model and significantly enhances its accuracy. The use of color in the heatmap clearly shows the correlation values between different variables. The correlation between water and cement usage is the most significant (R = 0.5), which is consistent with previous research findings.