## 4 Correlation Analysis.  
The Pearson correlation coefficient can be used to express the linear relationships between inputs and output, as shown in the heatmap of Figure 2. The color intensity in each square of the heatmap indicates the strength of the relationship between input and output, facilitating the easy identification of correlations. The values are represented with negative and positive signs to indicate inverse and direct relationships, respectively, on the heatmap. For example, the negative values and darker color intensity for C and Ni with experimental Ms highlight the more significant negative impact of these two elements on the Ms temperature. Similarly, the values close to zero and the very light color intensity for Mo in the heatmap indicate that it has a minimal effect on the Ms temperature.  
The heatmap analysis illustrates the correlation between alloying elements and their corresponding Ms temperature is not strictly linear, with significant differences. It reveals that there is not a substantial correlation between the elemental composition and the Ms temperature, suggesting a complicated, nonlinear relationship.  
Figure 1: Effect of various parameters on error for training, testing, and total datasets: (**a**) Number of neurons in a single hidden layer, (**b**) Number of neurons in two hidden layers, (**c**) Number of neurons in three hidden layers, (**d**) Learning rate, (**e**) Momentum term, (**f**) Number of iterations.  
Simulation and prediction using the trained ANN model.  
A JAVA-based virtual alloy addition system was developed for the ANN model. This system allowed for the straightforward prediction of the Ms temperature by inputting the alloying contents in steel. The graphical user interface (GUI), shown in Figure 3, clearly presents the correlation between the inputs (alloying content) and the output (predicted Ms temperature) in a simple and visual manner. This user interface represents a knowledge discovery in databases (KDD) model developed for predicting the Ms temperature of medium-carbon steels based on alloy compositions. The model is powered by an optimal ANN trained using precise synaptic weights to ensure high accuracy. By inputting the chemical composition of elements such as C, Mn, Si, Cr, Ni, Mo, V, Al, W, and Cu within the specified ranges, the model can predict Ms temperature with remarkable precision. The prediction output is dynamic, allowing for infinite combinations of compositions within the defined input ranges, making it a versatile tool for alloy design and research.  
In addition to predictions, this tool offers advanced functionalities such as one-variable and two-variable sensitivity analysis, enabling users to understand the influence of each element on Ms temperature. It also includes features to compare predicted outputs with training and test datasets to validate the model's accuracy. A description of the model, including its development methodology and features, is integrated into the interface for user understanding. Furthermore, the tool calculates the index of relative importance (I\({}_{\text{RI}}\)) for alloying elements, providing insight into their contribution to Ms temperature. Overall, this predictive model is a robust and user-friendly tool designed to support alloy development and metallurgical research.  
Figure 2: Colored heatmap illustrating the relationship between input and output variables using the Pearson correlation coefficient. The color gradient in each square represents the degree of correlation strength.