## 2 Materials and Methods
### Data and Methods Used in the Study  
We used the superconductivity dataset from [PERSON] [7], available from the UCI Machine Learning repository ([[https://archive.ics.uci.edu/ml/datasets/Superconductivy+Data](https://archive.ics.uci.edu/ml/datasets/Superconductivy+Data)]([https://archive.ics.uci.edu/ml/datasets/Superconductivy+Data](https://archive.ics.uci.edu/ml/datasets/Superconductivy+Data)), accessed on 11 September 2020). The website contains two files. In this work, we only used the _train.csv_ file, which contains information on 21,263 superconductors, including the critical temperature and a total of 81 attributes for each of them. This dataset on superconductors utilizes elemental properties to predict the critical temperature (\(T_{c}\)). The data extraction process involves obtaining ten features from the chemical formula for each of the eight variables, which include _Atomic Mass, First Ionization Energy, Atomic Radius, Density, Electron Affinity, Fusion Heat, Thermal Conductivity,_ and _Valence_. This results in a total of 80 features. Additionally, the dataset includes one extra feature representing the count of elements present in the superconductor. The dataset encompasses both \"oxides\" and \"metallic\" materials, but excludes elements with atomic numbers greater than 86. After performing thorough data preparation and cleaning steps, the final dataset consisted of 21,263 samples, each described by 81 features. Regarding the elemental distributions of the superconductors in the dataset, Oxygen constitutes approximately 56% of the total composition. Following Oxygen, the next most abundant elements are Copper, Barium, Strontium, and Calcium. As many of the research group's primary interest is in iron-based superconductors, the following information is likely to be of significant interest to them. Within this dataset, Iron is present in about 11% of the superconductors, with a mean critical temperature (\(T_{c}\)) of 26.9 \(\pm\) 21.4 K. On the other hand, the mean \(T_{c}\) for non-iron-containing superconductors is 35.4 \(\pm\) 35.4 K. Looking at the overall distribution of \(T_{c}\) values, it is found to be right-skewed, with a noticeable peak centered around 80 K.  
Figure 1: Examples of the fit obtained by the _Spline Continued Fraction_ using a dataset generated thanks to the gamma function with added noise. We present several continued fractions with depths of 3 (**a**), 5 (**b**), 10 (**c**), and 15 (**d**). In this example, the number of knots \(k\) was chosen to be 3, \(norm=1\), and \(\lambda=0.1\). For a more detailed understanding of the data generation, feature extraction process, and specific characteristics of the dataset, readers can refer to [PERSON]'s original contribution in [7]. We conducted two main studies to assess the generalization capabilities of many regression algorithms. We denote these as _Out-of-Sample_ and _Out-of-Domain_, respectively. For the _Out-of-Sample_ study, the data were randomly partitioned into two thirds training data and one third test data. Each model was fitted to the training data, and the _RMSE_ was calculated on the separated test portion of the data. For the _Out-of-Domain_ study, the data were partitioned such that the training samples were always extracted from the set of samples with the lowest 90% of critical temperatures. For the test set, the samples come from the highest 10% of critical temperatures. It turned out that the lowest 90% had critical temperatures < 89 K, whereas the highest 10% had temperatures greater than or equal to 89 K that ranged from 89 K to 185 K (we highlight that the range of variation of the test set is more than that of the training set, making the generalization task a challenging one). For each of the 100 repeated runs of the _Out-of-Domain_ test, we randomly took half of the training set (from lowest 90% of the observed value) to train the models and the same ratio from the test data (from 10% of the highest actual value) to estimate the model performance.