1 Introduction
2 Materials and methods
2.1 Dataset
Variable | Units | Description |
---|---|---|
Atomic Mass | Atomic mass units (AMU) | Total proton and neutron rest masses |
First Ionization Energy | Kilo-Joules per mole (kJ/mol) | Energy required to remove a valence electron |
Atomic Radius | Picometer (pm) | Calculated atomic radius |
Density | Kilograms per meters cubed (kg/m3) | Density at standard temperature and pressure |
Electron Affinity | Kilo-Joules per mole (kJ/mol) | Energy required to add an electron to a neutral atom |
Fusion Heat | Kilo-Joules per mole (kJ/mol) | Energy to change from solid to liquid without temperature change |
Thermal Conductivity | Watts per meter-Kelvin (W/(m K)) | Thermal conductivity coefficient κ |
Valence | No units | Typical number of chemical bonds formed by the element |
Feature and description | Formula | Sample value (Re7Zr1) |
---|---|---|
Mean | \(\mu = \left( {t_{1} + t_{2} } \right)/2\) | 35.5 |
Weighted mean | \(\nu = \left( {p_{1} t_{1} } \right) + \left( {p_{2} t_{2} } \right)\) | 39.67 |
Geometric mean | \(= \sqrt {t_{1} t_{2} }\) | 33.23 |
Weighted geometric mean | \(= \left( {t_{1} } \right)^{{p_{1} }} \left( {t_{2} } \right)^{{p_{2} }}\) | 37.56 |
Entropy | \(= - w_{1} \ln \left( {w_{1} } \right) - w_{2} \ln \left( {w_{2} } \right)\) | 0.63 |
Weighted entropy | \(= - A\ln \left( A \right) - B\ln \left( B \right)\) | 0.44 |
Range | \(= t_{1} - t_{2} \,\,\left( {t_{1} > t_{2} } \right)\) | 25 |
Weighted range | \(= p_{1} t_{1} - p_{2} t_{2}\) | 24.33 |
Standard deviation | \(= \left[ {\left( {1/2} \right)\left( {\left( {t_{1} - \mu } \right)^{2} + \left( {t_{2} - \mu } \right)^{2} } \right)} \right]^{\frac{1}{2}}\) | 12.5 |
Weighted standard deviation | \(= \left[ {p_{1} \left( {t_{1} - \nu } \right)^{2} + p_{2} \left( {t_{2} - \nu } \right)^{2} } \right]^{\frac{1}{2}}\) | 11.79 |
2.2 Multivariate adaptive regression splines (MARS) approach
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constant and equal to 1. This term is called intercept and corresponds to the term \(c_{0}\);
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a hinge or hockey stick function: this function is \(\max \left( {0,constant - x} \right)\) or \(\max \left( {0,x - {\text{constant}}} \right)\). The constant value is termed knot. The MARS technique chooses variables and knot values for these according to the procedure indicated later;
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the multiplication of hinge functions: in this case, these functions model nonlinear relationships between variables.
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Forward stage: MARS starts with the intercept term, which is calculated by averaging the values of the dependent variable. Next, it adds linear combinations of pairs of hinge functions with the aim of minimizing the least-square error. These new hinge functions depend on a knot and a variable. Thus, to add new terms MARS has to try all the different combinations of variables and knots with the previous terms, called parent terms. Then, the coefficients \(c_{i}\) are determined using linear regression. Finally, it adds terms until a certain threshold for the residual error or a maximum number of terms is reached.
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Backward stage: the previous stage usually constructs an overfitted model. In order to construct a better model with greater generalization skill, this new stage simplifies the model by removing terms, using the generalized cross-validation (GCV) criterion described below by first removing the terms that add more GCV to the model.
2.3 Whale optimization algorithm (WOA)
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Encircling prey
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Exploitation phase: bubble-net attack method
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Exploration phase: search for prey
2.4 Ridge regression (RR)
2.5 Least absolute shrinkage and selection operator (Lasso) regression (LR)
2.6 Elastic-net regression (ENR)
2.7 Approach accuracy
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\(SS_{tot} = \sum\nolimits_{i = 1}^{n} {\left( {t_{i} - \overline{t}} \right)^{2} }\): is the overall sum of squares, proportional to the sample variance.
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\(SS_{reg} = \sum\nolimits_{i = 1}^{n} {\left( {y_{i} - \overline{t}} \right)^{2} }\): is the regression sum of squares, also termed the explained sum of squares.
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\(SS_{err} = \sum\nolimits_{i = 1}^{n} {\left( {t_{i} - y_{i} } \right)^{2} }\): is the residual sum of squares.
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Maximum number of basis functions (Maxfuncs): maximum number of model terms before pruning, i.e., the maximum number of terms created by the forward pass.
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Penalty parameter (d): the generalized cross-validation (GCV) penalty per knot. A value of 0 penalizes only terms, not knots. The value \(- 1\) means no penalty.
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Interactions: maximum degree of interaction between variables.
MetaheuristicOpt
package [20, 52] from the R Project. Additionally, the Ridge, Lasso, and Elastic-net regression models were implemented by using the glmnet
package [55].MARS hyperparameters | Lower limit | Upper limit |
---|---|---|
Maximum number of basis functions (MaxFuncs) | 3 | 100 |
Interactions | 1 | 4 |
Penalty parameter (d) | −1 | 4 |
3 Analysis of results and discussion
Hyperparameters | Optimal values |
---|---|
MaxFuncs | 56 |
Interactions | 2 |
Penalty (d) | 1 |
\(B_{i}\) | Definition | \(c_{i}\) |
---|---|---|
\(B_{1}\) | 1 | \(8.2954365\) |
\(B_{2}\) | h(159-range_atomic_radius) | \(- 0.0310443\) |
\(B_{3}\) | h(range_atomic_radius-159) | \(0.1093384\) |
\(B_{4}\) | h(6889.5-mean_Density) | \(0.0056201\) |
\(B_{5}\) | h(wtd_std_ThermalConductivity-85.1085) | \(0.3020147\) |
\(B_{6}\) | h(54.1925-std_atomic_mass)\(\times\)h(wtd_std_ThermalConductivity-85.1085) | \(0.0014886\) |
\(B_{7}\) | h(std_atomic_mass-54.1925)\(\times\)h(wtd_std_ThermalConductivity-85.1085) | \(0.0263762\) |
\(B_{8}\) | h(64.2578-wtd_std_atomic_mass)\(\times\)h(wtd_std_ThermalConductivity-85.1085) | \(- 0.0080287\) |
\(B_{9}\) | h(wtd_std_atomic_mass-64.2578)\(\times\)h(wtd_std_ThermalConductivity-85.1085) | \(- 0.0379316\) |
\(B_{10}\) | h(310.6-range_fie)\(\times\)h(wtd_std_ThermalConductivity-85.1085) | \(0.0027402\) |
\(B_{11}\) | h(range_fie-310.6)\(\times\)h(wtd_std_ThermalConductivity-85.1085) | \(- 0.0001969\) |
\(B_{12}\) | h(range_atomic_radius-159)\(\times\)h(wtd_mean_Valence-2.26857) | \(- 0.0748170\) |
\(B_{13}\) | h(range_atomic_radius-159)\(\times\)h(2.26857-wtd_mean_Valence) | \(3.1894155\) |
\(B_{14}\) | h(6889.5-mean_Density)\(\times\)h(wtd_gmean_ThermalConductivity-6.09074) | \(- 0.0000326\) |
\(B_{15}\) | h(6889.5-mean_Density)\(\times\)h(6.09074-td_gmean_ThermalConductivity) | \(- 0.0009077\) |
\(B_{16}\) | h(2006.63-gmean_Density)\(\times\)h(wtd_std_ThermalConductivity-85.1085) | \(0.0002240\) |
\(B_{17}\) | h(gmean_Density-2006.63)\(\times\)h(wtd_std_ThermalConductivity-85.1085) | \(- 0.0000403\) |
\(B_{18}\) | h(79.0562-wtd_gmean_Density)\(\times\)h(wtd_std_ThermalConductivity-85.1085) | \(0.0078285\) |
\(B_{19}\) | h(wtd_gmean_Density-79.0562)\(\times\)h(wtd_std_ThermalConductivity-85.1085) | \(0.0000419\) |
\(B_{20}\) | h(46.9714-wtd_range_ElectronAffinity)\(\times\)h(wtd_std_ThermalConductivity-85.1085) | \(- 0.0138414\) |
\(B_{21}\) | h(wtd_range_ElectronAffinity-46.9714)\(\times\)h(wtd_std_ThermalConductivity-85.1085) | \(- 0.0016589\) |
\(B_{22}\) | h(60.1526-wtd_std_ElectronAffinity)\(\times\)h(wtd_std_ThermalConductivity-85.1085) | \(- 0.0153485\) |
\(B_{23}\) | h(wtd_std_ElectronAffinity-60.1526)\(\times\)h(wtd_std_ThermalConductivity-85.1085) | \(- 0.0071685\) |
\(B_{24}\) | h(8.6244-mean_FusionHeat)\(\times\)h(wtd_std_ThermalConductivity-85.1085) | \(- 0.0617544\) |
\(B_{25}\) | h(mean_FusionHeat-8.6244)\(\times\)h(wtd_std_ThermalConductivity-85.1085) | \(- 0.0069835\) |
\(B_{26}\) | h(0.534908-wtd_entropy_ThermalConductivity)\(\times\)h(wtd_std_ThermalConductivity-85.1085) | \(0.0865304\) |
\(B_{27}\) | h(wtd_entropy_ThermalConductivity-0.534908)\(\times\)h(wtd_std_ThermalConductivity-85.1085) | \(1.1084331\) |
\(B_{28}\) | h(wtd_std_ThermalConductivity-85.1085)\(\times\)h(wtd_mean_Valence-2.38385) | \(- 0.1650073\) |
\(B_{29}\) | h(wtd_std_ThermalConductivity-85.1085)\(\times\)h(2.38385-wtd_mean_Valence) | \(- 0.5945354\) |
\(B_{30}\) | h(wtd_std_ThermalConductivity-85.1085)\(\times\)(std_Valence-0.433013) | \(0.2547686\) |
\(B_{31}\) | h(wtd_std_ThermalConductivity-85.1085)\(\times\)h(0.433013-std_Valence) | \(- 0.8776123\) |
\(B_{32}\) | h(wtd_std_ThermalConductivity-85.1085)\(\times\)h(0.515047-wtd_std_Valence) | \(1.3096580\) |
Error measure | WOA/MARS | Ridge | Lasso | Elastic-net |
---|---|---|---|---|
R2 | 0.8005 | 0.6936 | 0.7295 | 0.7291 |
r | 0.8950 | 0.8334 | 0.8541 | 0.8539 |
RMSE | 15.14 | 18.77 | 17.64 | 17.65 |
MAE | 10.75 | 14.50 | 13.43 | 13.44 |
3.1 Significance of variables
Input variable | Nsubsets | GCV | RSS |
---|---|---|---|
wtd_std_ThermalConductivity | 31 | 100.0 | 100.0 |
std_atomic_mass | 30 | 57.9 | 58.0 |
range_atomic_radius | 30 | 57.9 | 58.0 |
wtd_mean_Valence | 30 | 57.9 | 58.0 |
gmean_Density | 29 | 42.0 | 42.2 |
wtd_entropy_ThermalConductivity | 28 | 32.8 | 33.0 |
wtd_std_ElectronAffinity | 27 | 27.2 | 27.5 |
mean_Density | 26 | 25.4 | 25.6 |
wtd_range_ElectronAffinity | 25 | 23.2 | 23.5 |
std_Valence | 24 | 22.1 | 22.4 |
wtd_gmean_ThermalConductivity | 23 | 21.0 | 21.3 |
wtd_std_Valence | 20 | 17.9 | 18.1 |
wtd_std_atomic_mass | 17 | 14.4 | 14.7 |
range_fie | 14 | 11.9 | 12.2 |
wtd_gmean_Density | 12 | 9.9 | 10.2 |
mean_FusionHeat | 11 | 8.7 | 9.0 |
Material | Observed Tc (°K) | Predicted Tc (°K) |
---|---|---|
Mg1B1.94C0.06 | 34.8 | 21 |
Y1 | 2.5 | 3 |
Sm1.25Gd0.6Ce0.15Cu1O3.97 | 16.8 | 18.6 |
Bi2Sr2Ca1Cu2O | 86.3 | 78.2 |
Li1.4Zr1N1Cl1 | 10 | 10 |
Gd1.1Ba1.9Cu3O7 | 93 | 80.1 |
Pb1Mo6S6O2 | 11.7 | 4.1 |
Zr41.2Ti13.8Cu12.5Ni10Be22.5 | 0.9 | 23.6 |
Hg1Ba2Cu0.99Mg0.01O4 | 94 | 82 |
Pd0.95Pt0.05Te2 | 1.71 | 4.4 |
Hg0.9Ba2Cu1.05O4 | 96 | 82.9 |
Sn0.05Fe1Se0.93 | 7 | 6 |
Sr1Fe1.75Rh0.25As2 | 21.9 | 9.1 |
Nb0.29Re0.71 | 5.6 | 3.8 |
Y1Ba1.6Sr0.4Cu3O6.8 | 88.5 | 70.7 |
Mo0.865Re0.135 | 6.1 | 3.5 |
Lu4Sc1Ir4Si10 | 6.64 | 7.8 |
Y0.8Pr0.2Ba23O7.6 | 66.5 | 69.8 |
Na1Fe0.99Co0.01As1 | 17.8 | 15.3 |
4 Conclusion
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Existing analytical models to predict the superconducting critical temperature Tc from the observed values are not accurate enough as they make too many simplifications of a highly nonlinear and complex problem. Consequently, the use of machine learning methods such as the novel hybrid WOA/MARS-based approach employed in this study offer the best option for making accurate estimations of the Tc from experimental samplings.
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The hypothesis that the identification of Tc can be determined with precision by employing a hybrid WOA/MARS-based approach in a wide variety of superconductors has been successfully validated here.
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The application of this MARS-based methodology to the complete experimental dataset belonging to the Tc resulted in a satisfactory coefficient of determination and correlation coefficient whose values were 0.8005 and 0.8950, respectively.
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The ranking according to the order of importance of the input variables entailed in the estimation of the Tc from experimental samplings in different superconductors has been established. Specifically, Weighted Standard Thermal Conductivity has been identified as the single most important factor in predicting critical temperature Tc. It is also important to note the successive order of importance, which is as follows: the Standard Atomic Mass, Atomic Range Radius, Weighted Mean Valence, Geometric Mean Density, Weighted Entropy Thermal Conductivity, Weighted Standard Electron Affinity, Mean Density, Weighted Range Electron Affinity, Standard Valence, Weighted Geometric Mean Thermal Conductivity, Weighted Standard Valence, Weighted Standard Atomic Mass, Range First Ionization Energy, Weighted Geometric Mean Density and Mean Fusion Heat in the obtained Tc outcome.
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The principal role of the accurate hyperparameter determination in the MARS-based methodology in relation to the regression performance carried out for critical temperature Tc has been established using the WOA optimizer.