## 2 Materials and Methods
### Analysis of the Dataset  
In this study, the prediction of the hardness of heat-treated steel based on chemical composition and Jominy distance is investigated. The relationship between the Jominy distance and hardness, often referred to as the hardenability curve, serves as the central aspect of the analysis. It is assumed that the experiments conform to ASTM A-255 standards [46] to ensure consistency and reliability in data collection. In particular, it is assumed that the heat treatment procedures for the steel specimens meet the optimum conditions specified in the ASTM guidelines [46] to ensure the consistency and reliability of the test setup. In addition, the grain size of the steel samples is standardized to a value of seven on the ASTM scale, which contributes to the uniformity and reproducibility of the heat treatment process [2]. The dataset contains data on measurements on the Rockwell hardness scale C at distances of 1.5, 3, 5, 7, 9, 11, 13, 15, 20, 25, 30, 40, and 50 mm. It also contains information on the percentage of mass fraction of concentration of the seven basic alloying elements found in the group of alloyed structural alloy steel, namely C, Mn, Si, Cr, Ni, Mo, and Cu. Table 1 outlines the ranges of predictor parameters used in modeling the hardenability of steels. In MATLAB, the _cpartition_ function is a versatile tool for creating data partitions that is often used in machine learning and statistical analysis tasks. A typical application of this function is holdout validation, where the dataset is partitioned into a training set and a test set (or holdout set). The syntax _cpartition(n, 'HoldOut', \(p\))_ generates a random, non-stratified partition for holdout validation on a dataset with \(n\) observations. Here, \(p\) represents the proportion of observations assigned to the test set, while the remaining data form the training set. This method enables straightforward and efficient data partitioning for training and evaluation purposes in machine learning workflows [29]. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Range (\% Mass)** & **C** & **Mn** & **Si** & **Cr** & **Ni** & **Mo** & **Cu** & **Jominy Distance (mm)** \\ \hline Minimum & 0.12 & 0.36 & 0.12 & 0.09 & 0.04 & 0.0098 & 0.07 & 1.50 \\ \hline Maximum & 0.70 & 1.40 & 0.41 & 1.92 & 2.739 & 0.43 & 0.34 & 50 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Ranges of predictor parameters used to model hardenability of steels. Table 2 presents the descriptive statistics of the training and test datasets. It provides a comprehensive insight into the characteristics of the datasets and illustrates their distribution patterns and variability. In it, we examine critical statistical parameters, namely the maximum, minimum, mean, and standard deviation of the percentage mass fractions of seven elemental constituents and the Jominy distance, an indicator of hardenability. In this way, we examine the spectrum, central tendency, and scattering tendencies of the alloy compositions in the training and test datasets. The goal of predictive hardenability modeling is to create a model based on the training dataset and then to apply this model to the test dataset. However, to achieve the best results, the training dataset must be a representative sample of the data we want to apply it to (i.e., the test dataset). Otherwise, our model will be inadequate at best or utterly useless at worst. The Kolmogorov-Smirnov test checks whether the distributions of the two samples, namely the training and test datasets, differ significantly. The null hypothesis is that the samples come from the same distribution [47]. Table 3 presents the results of the Kolmogorov-Smirnov test, which is used to assess the goodness of fit of the training and test datasets. To assess the similarity of the feature distributions between the training and test datasets, we performed Kolmogorov-Smirnov (KS) tests at a 5% significance level for each feature.