Elemental compositional modeling of magnetic ordering temperature for spinel ferrite magnetocaloric compounds using intelligent algorithms

Abstract

Spinel ferrite recently attracted attention for possible application in magnetic refrigeration due to its noticeable high magnetocaloric effect and tunable magnetic ordering temperature around room temperature. Being a magnetic semiconductor, the material has enjoyed wider application in different practical domains such as drug delivery, humidity sensor, photo-catalyst, high density data storage, magnetic resonance imaging and magnetic cooling among others. However, simplicity of its preparation and excellent cost effectiveness as compared to the existing magnetocaloric-based materials further contribute to its suitability for attaining magnetic cooling. Effective utilization of this material for magnetic cooling requires precise measurement of its magnetic ordering temperature (MOT) which requires laborious experimental procedures and sophisticated equipment. This work addresses the challenges by employing elemental compositions of spine ferrite in developing hybrid models for predicting MOT using hybrid genetic-based support vector regression algorithm (GBSVRA) and extreme learning machine (ELM). The developed ELM-SN model with sine activation function performs better than hybrid GBSVRA and ELM-SG (with sigmoid activation function) model with performance improvement of 42.63% and 38.78%, respectively, through RMSE performance yardstick, while the ELM-SG model outperforms hybrid GBSVRA model with performance enhancement of 2.87% when validated using testing dataset. The developed ELM-SN model further outperforms other two developed models using other performance metrics. Harnessing the potentials of the presented models would strengthen precise, effective and quick tuning of spinel ferrite MOT for achieving magnetic cooling without experimental cost and difficulties.

Keywords:

• Extreme learning machine
• spinel ferrite
• magnetic ordering temperature
• support vector regression
• ionic radii
• genetic algorithm

1. Introduction

Spinel ferrites are magnetic oxide materials with distinct and attractive features such as high DC resistivity, high temperature chemical stability, reduced cost, mechanical hardness, low porosity, synthetic simplicity, high permeability and high magnetic ordering temperature (S. Hcini et al., 2018). These features have significantly attracted attentions within scientific community and subsequently aid the industrial as well as technological applications of spinel ferrite materials in memory devices, transformer cores, magnetic sensors, satellite communication, gas sensors, computer components, enhancement of contrast in magnetic resonance imaging and magnetic refrigeration applications, among others (Masina et al., 2015). Aside from large magnetocaloric effect and excellent relative cooling power which are essential ingredients for magnetic refrigeration application, adjustment of magnetic ordering temperature around room temperature is highly essential for this application (Hamad et al., 2021). Magnetic refrigeration is a novel technology based on magnetocaloric effect which has excellent promising features in replacing the conventional gas system of refrigeration (Owolabi, Akande, Olatunji, Aldhafferi, Alqahtani et al., 2019a, 2019b). This refrigeration method saves energy, is highly efficient and is economically clean (Chi Linh et al., 2018). Magnetocaloric effect in spinel ferrite materials occurs when the order of magnetic moments of the ferrite changes due to the change in the applied magnetic fields. The orderliness of magnetic moment in spinel ferrite magnetic materials increases as the applied external field increases and becomes disordered when the applied field is reduced or removed (Zhao et al., 2021). The variation in the orderliness of spinel ferrite magnetic materials results in cooling since heat is absorbed by the system for energy increase. The magnetic ordering temperature (MOT) at which the ordering and disordering is instituted should fall within room temperature for effective cooling process. Adjusting MOT of this material requires laborious experimental procedures and sophisticated equipment. In order to facilitate quick and precise prediction of MOT for spine ferrite compounds, this work develops hybrid genetic-based support vector regression algorithm (GBSVRA) and extreme learning machine (ELM) using elemental compositions of spinel ferrite as descriptors.

Spinel ferrite magnetic materials are of general chemical formula AB₂O₄ in which A and B respectively indicate the sites for tetrahedral and octahedral ions (Bahhar et al., 2021). A-site can be occupied by divalent cations, while trivalent cations occupy B-site. In accordance to cation distribution between A-sites and B-sites as interlinked with the degree of inversion and divalent metallic ions lattice occupancy in crystallographic sites, spinel ferrite is categorized into inverse spinel (zero degree of inversion), normal spinel (degree of inversion is one) and mixed spinel (degree of inversion is 0.25; Sagar et al., 2019). Spinel ferrite materials have characteristic face-centered cubic structure with crystallographic sub-lattices merged between octahedral and tetrahedral with antiparallel spin orientation. The magnetic and other physical properties of spinel ferrite are influenced through cation distribution among tetrahedral and octahedral sites which is consequent upon stoichiometry, valency and ionic occupancy (Bouhbou et al., 2022; Felhi et al., 2018). MOT is the thermal point beyond which magnetic characteristics of spinel ferrite are lost (Bahhar et al., 2021). The super-exchange interactions within material’s magnetic atoms are controlled by MOT. Coupling between 3d-3d divalent and trivalent cations positioned at tetrahedral and octahedral sites controls ferrite magnetic behavior. Substitution of external magnetic materials such as rare earth elements leads to the formation of new magnetic exchange interactions through indirect coupling (4 f-5d-5d-4 f) and direct exchange coupling (3d-4 f). This substitution has potential to weaken or strengthen the intra- and inter-sublattice super-exchange interactions depending on the magnetic moment strength of the dopants. Hence, magnetic ordering temperature can thereby be influenced (Bahhar et al., 2021). Increase in the concentration of non-magnetic ions such as Zn²⁺ pushes Fe³⁺ ions from A-site to B-site and thereby leads to decrease in tetrahedral and octahedral interaction which ultimately decreases the MOT (Anwar et al., 2014). The MOT of spinel ferrite magnetic materials is modeled in this contribution through elemental composition and stoichiometry of the materials constitutional elements using intelligent algorithms.

Support vector regression is a supervised learning algorithm that addresses real-life problems through regression function approximation using the principle of structural risk minimization (Vapnik, 1995). Existing soft computing techniques of addressing real-life problems include fuzzy knowledge-based system (Sujatha et al., 2021), support direction vector-based model (J. Prottasha et al., 2021) and sensitivity-based linear learning method (Owolabi & Gondal, 2015), among others (Mohammed et al., 2022) . However, support vector regression-based model has been reported to be effective while handling problem with characteristic few dataset and the algorithm sets a balance between model’s complexity (through margin maximization) and training dataset fitting. Margin maximization ensures optimization of the distance between the samples that is closest to the boundary which partitioned the training patterns (Science et al., 2021). Euclidean norm minimization that translates to decision function flatness through the associated weight vector with the decision boundary is controlled through the user-defined hyper-parameter known as the penalty factor (Owolabi, 2019a). Support vectors (which serve as the model for future implementation) generation involves quadratic programming problem solving with the sole aim of setting a balance between two main objective functions (Pablo et al., 2021). These objectives include the construction of linear regressor through model fitness maximization and complexity reduction. Support vector regression algorithm has been extensively utilized in addressing numerous problems due to its aforementioned unique features and characteristics (Durgam et al., 2020; Oloore et al., 2018; Zhang et al., 2015). This work optimizes the user-defined hyper-parameters of support vector regression through genetic algorithm hybridization. This optimization technique has potential features for the precision and accuracy enhancement when hybridized with support vector regression-based model due to its avoidance of premature convergence and attainment of global solution within shortest period of computational time (Owolabi, 2019b).

The conventional neural network-based feed-forward trains the network through gradient descent method which is characterized with certain limitations (L. Li et al., 2021). This training method allows setting of a number of parameters during the training process while iterative procedures are employed for adjusting the network weights. Hence, the model is characterized with slow training process. Extreme learning machine (ELM) addresses these challenges through presence of single hidden layer and random generation of input weights as well as the biases (Bin Huang et al., 2006; Owolabi, 2020; Oyeneyin et al., 2021). This uniqueness has promoted and necessitated wider applications of ELM-based algorithm in various fields (Alqahtani, 2021; Owolabi & Gondal, 2018; Oyeneyin et al., 2022) . Therefore, ELM trains the network at a fast speed and effectively attains universal approximation of the patterns linking the descriptors with the desired target. These merits are utilized in this work in modeling MOT of spinel ferrite magnetocaloric compound using ionic radii and elemental concentrations as model descriptors.

The organization of the rest of the manuscript goes thus: section two discusses the mathematical framework of extreme learning machine, genetic algorithm and support vector regression. Section three presents the computational implementation of the proposed hybrid model as well as extreme learning machine. Section four discusses the outcomes of each of the model with their performance comparison. The conclusions drawn from the research work are presented in the last section of the manuscript.

2. Mathematical framework of the employed algorithms

The mathematical frameworks of support vector regression are presented this section. Principles of operations genetic algorithm are also described in this section.

2.1. Support vector regression intelligent algorithm

Machine learning support vector regression is a class of algorithms designed for solving real-life problems with characteristic non-linearity, small data sample and high dimensionality (Li et al., 2022; Olubosede et al., 2022; Vapnik, 1998). Consider spinel ferrite samples $\mathit{S}=\left({\lambda }_j,T_{\mathit{C}\mathrm{\_}{exp}_j}\right)$ $\mathit{j}=,2,\dots .,\mathit{k}$ in which ${\lambda }_j$ represents the input vector (elemental compositions of spinel ferrite compound which include the ionic radii of each of the constituents as well as their concentration) while $T_{\mathit{C}\mathrm{\_}{exp}_j}$ stands for the measured magnetic ordering temperature of the compounds and k is the number of modeling samples. Mapping functions such as sigmoid, polynomial and Gaussian are frequently employed for high dimensional space sample mapping $\left(\mathit{\lambda }\to \mathit{\alpha }(\mathit{\lambda })\right)$. The algorithm employs the feature space dot product $\mathrm{RSCG\_A\_}2172859$ during optimal hyper-plane construction. Therefore, a function that satisfies $\mathit{\beta }({\lambda }_j,{\lambda }_i)=\mathit{\alpha }({\lambda }_j).\mathit{\alpha }({\lambda }_i)$ is the kernel function that helps in data sample mapping (Fan et al., 2021). Equation (1)(1) $\mathit{\beta }({\lambda }_j,{\lambda }_i)=exp\left(\frac{-({\lambda }_j-{\lambda }_i)}{2\mathit{\rho }}\right)$(1) presents the optimum kernel function for magnetic ordering temperature modeling presented in this work.

(1) $\mathit{\beta }({\lambda }_j,{\lambda }_i)=exp\left(\frac{-({\lambda }_j-{\lambda }_i)}{2\mathit{\rho }}\right)$(1)

where $\mathit{\rho }$ is the kernel parameter.

Although support vector-based algorithm was initially designed for solving classification problems. However, inclusion of kernel trick and other mechanisms elevates the algorithm to address non-linear and linear regression problems. The linear function governing SVR-based model is presented in equation (2)(2) $T_{\mathit{C}\mathrm{\_pred}}=\mathit{\omega }.\mathit{\lambda }+\mathit{\varphi }$(2) .

(2) $T_{\mathit{C}\mathrm{\_pred}}=\mathit{\omega }.\mathit{\lambda }+\mathit{\varphi }$(2)

where $\mathit{\varphi }$ = bias (regression function’s offsets), $\mathrm{RSCG\_A\_}2172859$ = predicted magnetic ordering temperature and $\mathit{\omega }$ = weight (normal) vector.

Small value of $\mathit{\omega }$ ensures flatness of equation (2)(2) $T_{\mathit{C}\mathrm{\_pred}}=\mathit{\omega }.\mathit{\lambda }+\mathit{\varphi }$(2) which calls for minimization of its norm within Euclidean space. With the assumption of fitting the training data into a linear function after or before transformation to space of high dimension (depending on the nature of the problem) with maximum error threshold of $\mathit{\epsilon }$, optimization problem presented in equation (3)(3) $\begin{array}{c}\mathrm{minimization}\mathrm{of}\frac{{\mathit{\omega }}^2}{2}\\ \mathrm{subjected}\mathrm{to}:\\ T_{\mathit{C}\mathrm{\_}{exp}_j}-\mathit{\omega }.{\lambda }_j-\mathit{\varphi }\le \mathit{\epsilon }\\ \mathit{\omega }.{\lambda }_j+\mathit{\varphi }-T_{\mathit{C}\mathrm{\_}{exp}_j}\le \mathit{\epsilon },\mathit{j}=1,2,..,\mathit{k}\end{array}$(3) needs to be addressed.

(3) $\begin{array}{c}\mathrm{minimization}\mathrm{of}\frac{{\mathit{\omega }}^2}{2}\\ \mathrm{subjected}\mathrm{to}:\\ T_{\mathit{C}\mathrm{\_}{exp}_j}-\mathit{\omega }.{\lambda }_j-\mathit{\varphi }\le \mathit{\epsilon }\\ \mathit{\omega }.{\lambda }_j+\mathit{\varphi }-T_{\mathit{C}\mathrm{\_}{exp}_j}\le \mathit{\epsilon },\mathit{j}=1,2,..,\mathit{k}\end{array}$(3)

Relaxation variables known as slack variables (${\gamma }_j$ and ${\gamma }_j^{\ast }$) are introduced when the constraints contained in equation (3)(3) $\begin{array}{c}\mathrm{minimization}\mathrm{of}\frac{{\mathit{\omega }}^2}{2}\\ \mathrm{subjected}\mathrm{to}:\\ T_{\mathit{C}\mathrm{\_}{exp}_j}-\mathit{\omega }.{\lambda }_j-\mathit{\varphi }\le \mathit{\epsilon }\\ \mathit{\omega }.{\lambda }_j+\mathit{\varphi }-T_{\mathit{C}\mathrm{\_}{exp}_j}\le \mathit{\epsilon },\mathit{j}=1,2,..,\mathit{k}\end{array}$(3) are not satisfied. The convex optimization problem is then transformed to the expression contained in equation (4).

(4) $\left\{\begin{array}{c}\mathrm{minimization}\mathrm{of}\mathit{\sigma }(\mathit{\omega })=\frac{{\mathit{\omega }}^2}{2}+\mathit{\mu }\sum _{\mathit{j}=1}^k\left({\gamma }_j^{\ast }+{\gamma }_j\right)\\ \mathrm{subjected}\mathrm{to}:\\ T_{\mathit{C}\mathrm{\_}{exp}_j}-\mathit{\omega }.{\lambda }_j-\mathit{\varphi }\le \mathit{\epsilon }+{\gamma }_j\\ \mathit{\omega }.{\lambda }_j+\mathit{\varphi }-T_{\mathit{C}\mathrm{\_}{exp}_j}\le \mathit{\epsilon }+{\gamma }_j^{\ast },({\gamma }_j^{\ast },{\gamma }_j\ge 0),\mathit{j}=1,2,..,\mathit{k}\end{array}\right.$(4)

where $\mathit{\mu }$ is the penalty factor.

The quadratic programming problem (since $\mathit{\sigma }(\mathit{\omega })$ is quadratic in nature) is well solved through Lagrange multipliers method (Owolabi & Gondal, 2015). The final decision function for predicting the magnetic ordering temperature of spinel ferrite materials is presented in equation (5)(5) $T_{\mathit{C}\mathrm{\_}exp}=\sum _{\mathit{j}=1}^k\left({\tau }_j-{\tau }_j^{\ast }\right)\mathit{\beta }\left({\lambda }_j,{\lambda }_i\right)+\mathit{\varphi }$(5) with Lagrange multipliers ${\tau }_j$ and ${\tau }_j^{\ast }$

(5) $T_{\mathit{C}\mathrm{\_}exp}=\sum _{\mathit{j}=1}^k\left({\tau }_j-{\tau }_j^{\ast }\right)\mathit{\beta }\left({\lambda }_j,{\lambda }_i\right)+\mathit{\varphi }$(5)

The penalty factor $\mathit{\mu }$, maximum error threshold $\mathit{\epsilon }$ and kernel parameter $\mathit{\rho }$ are optimized and tuned through genetic algorithm heuristic optimization method.

2.2. Genetic algorithm

Genetic algorithm is a method of optimization that simulates the Darwin natural evolution process and employs genetic vocabularies to address more complex natural optimization system (Holland, 1992). The algorithm has intrinsic advantageous features that include its potentials in managing mixed variables (continuous, discrete and categorical), wide possible search space, global convergence and multiple optimal solutions instead of a single solution. These characteristics further strengthen wider applicability of the algorithm for addressing real-life challenges. In optimization description of genetic natural evolution process, individuals in a given population are made up of chromosomes which, in turn, consist of genes that carry distinct inheritance features of the individual (Z. X. Li et al., 2021). The genetic natural process inspires evolution operations which include selection, crossover and mutation which are probabilistically controlled by a set of values. Two variants are involved in selection operation which includes the parental selection (for reproduction purpose) and replacement selection (for the purpose of maintaining constant population size). The selection process prevents premature convergence by giving chance to less privileged individuals after consideration of the best individuals (Dhagat & Jujjavarapu, 2021). The crossover operation explores the research space further and ensures population diversification through the creation of new offsprings by combining intricacies and information contained in two parents. The crossover operation might be two-points, one-points or uniform crossover operation (Gharsalli & Guérin, 2021). Individual’s chromosome undergoes slight perturbation through mutation operation and subsequently leads to avoidance of local solution convergence through gene diversity enhancement.

3. Algorithms hybridization and computational strategies

Acquisition details of the employed set of data are presented in this section. Computational methodologies of the hybrid genetic and support vector regression are contained in this part of the manuscript.

3.1. Dataset description and acquisition

Twenty-nine spinel ferrite magnetocaloric samples with different dispersion of cations among octahedral and tetrahedral sites were employed in modeling the MOT using two intelligent algorithms. The experimental values of MOT for each of the samples were drawn from the literature (Bahhar et al., 2021; Bouhbou et al., 2022, 2017; Felhi et al., 2018; Fortas et al., 2020; F. Hcini et al., 2021; Oumezzine et al., 2015; Zhao et al., 2021). The descriptors to the developed models are the ionic radii of the compound constituent elements and their respective concentration. The expression governing the operational implementation of the proposed models for predicting MOT of spinel ferrite magnetocaloric compound is presented in equation (6)(6) $A_x{C}_y{D}_z\mathit{F}{e}_2{O}_4$(6) .

(6) $A_x{C}_y{D}_z\mathit{F}{e}_2{O}_4$(6)

The position of divalent ions for tetrahedral sites is represented by A while C and D are the possible dopants for improving MOT and other magnetic properties spinel ferrite. The ionic radius of divalent ions A, the ionic radius of dopant C and D and their concentrations x, y and z are the descriptors to the model. The value of applied external magnetic field (H) also serves as the descriptor to the models. For example, if proposed models are to be utilized for determining MOT of Zn₀₄Ni_0.2Cu_0.4Fe₂O₄ compound, the inputs to the models are the ionic radius of zinc (A), ionic radius of nickel (C), ionic radius of copper (D) and their respective concentrations of 0.4, 0.2 and 0.4. Similarly, implementation of the developed models for estimating MOT of Zn₀₆Cu_0.4Fe₂O₄ compound needs ionic radius of zinc (A), ionic radius of copper (D) and their respective concentration of 0.6 and 0.4, while the ionic radius of dopant D and its concentration are assigned zero values because of the absence of the dopant in the chemical structure. This gives advantage of incorporating two different dopants into spinel ferrite magnetocaloric compound that its MOT is desired. Analysis of the entire data sample is presented in Table 1.

Table 1. Analysis of data sample

Data sample contents	Mean	Standard deviation	Minimum	Maximum	Correlation coefficient
MOT	435.1034	181.1642	127.0000	705.0000	1.0000
A (pm)	84.0517	5.3358	68.5000	88.0000	−0.2128
x	0.4681	0.2184	0.2000	1.0000	0.4108
C (pm)	30.6552	43.4283	0.0000	109.0000	0.4874
y	0.0784	0.1210	0.0000	0.4000	0.5571
D (pm)	89.1379	7.2837	83.0000	115.0000	0.2460
z	0.5224	0.2378	0.1000	1.0000	−0.3293
H(T)	3.5414	1.5790	1.3000	5.0000	0.5566

The mean values of each of the descriptors, standard deviation, minimum, maximum and coefficients of correlation are presented in Table 1. Inferable information from the presented results of statistical analysis include the entire content of dataset (as observed from the mean values), consistency in the employed set of data (as observed from the value of standard deviation), range of the dataset (from maximum and minimum values) and the degree of linearity between the descriptors and the desired values of MOT (as inferred from the coefficient of correlation). Deficiency of linear models in establishing a universal relation between the elemental compositions of spinel ferrite compound and MOT is inferred from the values of coefficients of correlation. This necessitates the need to employ non-linear intelligent models proposed in this work.

3.2. Computational framework of support vector regression-based hybrid model

Genetic evolutionary algorithm is hybridized with support vector regression for developing a robust model (GBSVRA) through which MOT of magnetocaloric spinel ferrite compounds is estimated using elemental compositions of the magnetic compounds. The computational task for model development was implemented within computing environment of MATLAB. Samples from twenty-nine spinel ferrite magnetocaloric compounds available for simulation and modeling were initially randomized and subsequently separated into training and testing set in the ratio of 7:3. Dataset from twenty-one spinel ferrite samples was involved in support vector acquisition while eight spinel ferrite compounds were utilized for model validation. Aside from the computational efficiency invoked by randomization process, the possibility of overfitting can be nipped in the bud through an effective randomization process. Details of the algorithm hybridization for penalty factor, epsilon and kernel parameter optimization are presented as follows:

Step 1: Initialization of genetic algorithm parameters: The number of population size (20, 50,100 and 200), maximum number of generation (100), crossover probability (0.9) and mutation probability (0.005) were initialized within the search space. The search spaces for penalty factor (S. Hcini et al., 2018 −600), epsilon [0.001–0.009] and kernel parameter [0.1–0.3] were also defined and initiated.

Step 2: Fitness evaluation and computation: possibility of each of the chromosomes to evolve to next generation was evaluated after initial population generation through fitness computation. At this computational stage, the genetic algorithm was hybridized with support vector regression for fitness evaluation. The evaluation goes thus: (i) Choice of kernel function from Gaussian, polynomial and sigmoid (ii) combination of a chromosome from the initially generated population with the chosen kernel function and training dataset to train SVR algorithm (iii) comparison of the estimates of the trained algorithm from Step (ii) with the measured MOT for performance metrics determination. Training root mean square error (TRMSE) was noted with the corresponding support vectors. (iv) Combination of support vectors obtained in Step (iii) with testing data samples for determining the estimates of the model during testing phase. Determination of mode performance using testing root mean square error (TSRMSE) from model’s estimates (v) repeat Step (i) to Step (iv) for every chromosomes until all individual in the population is evaluated. (vi) Compare and rank the fitness of each of the chromosomes on the basis of TSRMSE with the convention that the lower the value of TSRMSE, the better and more fit the chromosomes for translation to the next stage of algorithm process.

Step 3: Selection operation for population replacement: Excellent chromosomes and individual parents are selected through selection operation with selection probability of 0.8. The two variants for the selection operation include the parental selection for reproduction purpose and replacement selection for maintaining constant population size.

Step 4: Crossover operation: this operation diversifies the population and further explores the search space for global convergence through formation of new offsprings from two parents by exchanging genes. The probability was set at 0.9 for optimum subsequence exchange.

Step 5: Mutation operation: Random changes are easily transferred between the parents and offsprings through mutation operation. Mutation also inverts some genes purposely to diversity the population. The probability was set at 0.005 so as to present random alteration of excellent chromosomes in the previous population.

Step 6: Stopping criteria: the algorithm is brought to stop after attaining maximum set iteration or same value of TSRMSE is obtained consecutively after for fifty iterations.

Step 7: Development of the final model: with the aid of saved support vectors to develop SVR model through estimation of MOT is conducted. Figure 1 presents the computational flowchart of the hybridized support vector regression and genetic algorithm.

Figure 1. Computational details of hybrid support vector regression and genetic algorithm.

4. Results and discussion

The outcomes of both hybrid intelligent model and ELM-based modes are presented here. The section also contains the performance comparisons between the models. Significance of different ions on MOT of spinel ferrite magnetocaloric compounds is contained here.

4.1. Combinatory choice of SVR parameters

The results of optimal selection of user-defined SVR parameters as obtained from optimization algorithm are shown in Figure 2. The regularization factor (that controls and balances the complexities of the model and the tendency of threshold error going beyond the set limit) is shown in Figure 2 (a) for varying number of population sizes. The regularization (penalty) factor convergence jumps to a higher value with the population size increment from twenty to fifty. Euclidean norm minimization that translates to decision function flatness through the associated weight vector with the decision boundary is controlled using regularization factor. Lower value of penalty factor does not justify precise model while this parameter is to be optimally selected to prevent overfitting of the model. The observed insignificant impact of the population size on the penalty factor has also been reported elsewhere (Owolabi et al., 2021) .

Figure 2. Combinatory influence of the utilized genetic algorithm (a) penalty factor at various iteration (b) Epsilon as a function of iteration (c) kernel mapping parameter as number of iteration progresses (d) error convergence at different iteration.

The convergence of epsilon error threshold of GBSVRA model is presented in Figure 2 (b) for a range of iteration at four different population sizes. The robustness in the model is inferred from the obtained universal convergence for various sizes of the chromosomes in the population. Convergence of the parameter that governs mapping of dataset to feature space for Gaussian kernel function is presented in Figure 2(c). Variation in the chromosome size in the search space has little significance on dataset mapping to feature space of high dimension. In the same vein, the performance of GBSVRA model for different number of chromosomes is shown in Figure 2 (d). The developed model demonstrates similar and robust convergence for various numbers of exploring and exploiting chromosomes in the search space. The influence of population size on the performance of the model depends on the nature of the problem and the data used for modeling and simulation. In the addressed problem in this case, population size of chromosome has little influence on epsilon error threshold and root mean square error. This similar behavior has also been reported elsewhere (Owolabi et al., 2021) . Table 2 presents the optimal values of GBSVRA parameters as extracted from the results of genetic algorithm.

Table 2. Results of the genetic algorithm (SVR parameters)

Tuning parameters in SVR	Global value
Kernel parameter	0.2107
Penalty factor	553.796
Kernel function	Gaussian
Epsilon	0.001
Population size	20
Hyper-plane lambda	0.0000001

4.2. Extreme learning machine approximated function

Mathematical expression for determining MOT of spinel ferrite magnetocaloric compounds as obtained from extreme learning machine intelligent algorithm is presented in equation (17)(17) $\mathit{MOT}=\sum _{\mathit{j}=1}^p{\delta }_j{g}_{\mathit{act}}\left({\chi }_j\mathit{\lambda }+{\eta }_j\right)$(17) .

(17) $\mathit{MOT}=\sum _{\mathit{j}=1}^p{\delta }_j{g}_{\mathit{act}}\left({\chi }_j\mathit{\lambda }+{\eta }_j\right)$(17)

where $g_{\mathit{act}}$ =activation function = sine function, ${\eta }_j$=biases (generated randomly at the commencement of the simulation), ${\chi }_j$ =input weight (randomly generated through Mersenne Twister generator), ${\delta }_j$ =output weight and $\mathit{p}$ =neuron number in hidden layer . The activation function is a sine function while the optimum number of neurons in the hidden layer is sixty-two as shown in the table. The needed weights for model reproducibility are shown in Table 3. The weight of ELM-SG model (with sigmoid activation function) is not presented while only the best ELM-based model is presented. The interesting feature of ELM-based model as compared with other intelligent algorithms such as SVR-based model is the simplicity of implementation for practical deployment. With the known weights of ELM-based model as shown in Table 3, the model can easily be implemented on Excel or even a calculator. The computational flowchart showing the modeling details of the developed ELM model is presented in Figure 3.

Figure 3. Computational flowchart of ELM-based model for MOT prediction.

Table 3. Extreme learning machine weights for model reproducibility

$\mathit{p}$	${\mathit{\chi }}_1$	${\mathit{\chi }}_2$	${\mathit{\chi }}_3$	${\mathit{\chi }}_4$	${\mathit{\chi }}_5$	${\mathit{\chi }}_6$	${\mathit{\chi }}_7$	$\mathit{\eta }$	$\mathit{\delta }$
1	−0.51442	0.151525	−0.43281	−0.86704	0.55957	−0.5095	0.777457	0.223514	76.4672
2	−0.80917	−0.45068	0.718078	−0.12048	0.374293	−0.53945	0.031748	0.541308	20.52116
3	−0.41783	0.341106	0.225585	0.208956	0.663016	0.774093	−0.67649	0.236643	−15.3892
4	−0.38577	0.633167	−0.19743	0.957932	−0.34641	0.78404	0.770397	0.906057	120.7967
5	−0.47773	−0.52775	−0.48743	−0.57335	−0.1234	−0.4297	−0.12157	0.941593	−98.9796
6	−0.59868	0.320187	−0.70865	0.070804	0.107112	−0.24257	0.356221	0.776397	−60.2309
7	−0.79962	−0.20243	0.577352	−0.56484	0.350181	−0.86789	−0.6626	0.264603	−18.4983
8	0.270824	−0.16955	0.3856	0.575908	−0.1193	−0.24123	−0.45716	0.195132	−75.3344
9	0.340729	−0.37352	0.654177	−0.86508	−0.96217	0.295099	−0.0482	0.58689	195.644
10	−0.07051	−0.9425	−0.93752	0.634082	0.955,957	0.80182	−0.01961	0.12487	−75.8461
11	0.369909	−0.18364	0.378473	−0.03254	−0.50891	0.102341	−0.39589	0.104899	−105.307
12	−0.77867	−0.7751	0.832865	0.529901	0.146851	0.994273	0.024252	0.793454	227.9922
13	0.821016	0.85932	0.964453	0.361826	−0.69734	−0.96171	−0.04577	0.992594	−24.3423
14	−0.08973	0.299174	−0.51289	−0.64904	−0.64072	0.507575	−0.21206	0.909765	−23.0427
15	−0.92027	0.999584	0.383714	0.869967	0.58757	−0.56553	−0.60631	0.54698	41.32443
16	−0.04494	−0.74659	0.165421	−0.45317	0.098467	0.518194	−0.6411	0.015117	73.46305
17	0.622783	−0.37899	−0.07266	−0.46926	0.663773	−0.3549	−0.68931	0.997319	−88.6559
18	−0.78973	0.070096	0.410547	−0.35925	0.452651	0.131605	0.70523	0.768356	15.5778
19	−0.38237	0.647427	−0.15006	0.653853	−0.61099	−0.37736	0.012908	0.013378	−134.387
20	0.977435	−0.81774	0.666434	0.630454	−0.93585	0.483203	0.633249	0.522898	−126.741
21	0.313783	0.010106	0.283035	0.515528	0.926921	0.262038	0.825067	0.12139	−54.4741
22	−0.60014	−0.67071	0.879881	−0.8095	0.587198	0.300808	−0.25754	0.263307	46.22696
23	−0.17224	0.384147	0.727237	−0.57865	−0.5857	0.06782	−0.13936	0.958459	−152.68
24	−0.22987	−0.79605	−0.20304	0.606628	0.822933	0.447162	0.295455	0.760732	60.34544
25	−0.45422	−0.04456	0.394616	0.060541	0.53102	−0.42409	−0.95641	0.468824	−4.36292
26	0.284952	−0.83335	0.739757	0.59883	−0.69984	−0.26852	−0.03739	0.247054	−13.5968
27	0.940503	−0.78427	−0.77965	0.189996	0.840038	−0.35564	0.804232	0.09899	−66.2988
28	0.208398	0.958938	0.072635	−0.1644	−0.79962	−0.01393	−0.73616	0.201595	113.3039
29	0.141952	−0.82512	0.140164	0.44286	−0.08406	−0.00586	−0.70423	0.971506	11.02041
30	0.212846	−0.85562	−0.06854	0.694439	−0.22971	0.962336	0.759488	0.814683	−279.042
31	−0.41665	−0.25776	−0.9756	0.447226	0.325068	−0.39841	0.287847	0.36057	89.63651
32	0.853431	0.95307	−0.58287	0.338494	0.579737	0.954724	0.366302	0.738951	−54.9099
33	−0.8989	−0.75724	0.265736	−0.07829	0.398651	0.720636	0.671711	0.470602	83.22384
34	0.116969	−0.97595	−0.19522	−0.2985	−0.30889	−0.84822	0.825154	0.836717	−49.9442
35	−0.83074	0.875486	−0.88185	0.61555	0.523008	0.320069	−0.01012	0.837285	22.48595
36	−0.08125	−0.66036	−0.07005	0.513692	0.20266	0.143828	0.865389	0.306878	5.458474
37	0.970573	−0.85927	0.084475	−0.1649	0.206746	0.926972	0.21228	0.138601	−99.1811
38	−0.29104	0.8238	0.260729	0.046355	−0.10938	0.183163	−0.33061	0.293205	−49.9669
39	0.117424	−0.7014	0.964604	−0.28035	0.507089	0.602329	−0.50693	0.498643	340.9157
40	−0.86686	−0.88254	−0.85346	−0.42886	−0.34353	0.412891	−0.29265	0.017486	−81.3404
41	0.746414	−0.92241	0.490665	0.288544	0.520277	0.979305	−0.71502	0.416498	−149.643
42	−0.53127	−0.1721	0.271789	−0.40229	−0.86941	0.03631	0.385583	0.078356	−38.6198
43	−0.64338	0.787192	−0.69247	−0.73417	−0.07329	−0.01149	0.325306	0.409607	−29.9596
44	−0.83909	0.313927	0.721478	0.449674	−0.13722	−0.21391	0.13829	0.249443	−40.6912
45	0.55663	0.23352	−0.65654	0.032435	−0.86543	−0.44689	−0.68503	0.356767	55.05174
46	−0.45897	−0.01062	−0.16238	0.646557	0.550094	−0.98171	−0.06412	0.893158	−111.963
47	0.558745	0.86732	0.444558	0.307026	−0.12374	0.944949	0.446797	0.001712	139.8847
48	−0.47612	−0.12164	0.923099	−0.37132	0.433339	−0.14079	−0.59591	0.532258	−51.384
49	0.235091	−0.17192	0.200231	−0.44194	0.026407	−0.40198	−0.59422	0.49563	77.20547
50	−0.98094	−0.49147	0.705473	−0.84763	0.336172	0.380735	0.44793	0.284686	26.19464
51	−0.68787	0.855684	0.412611	−0.67918	0.869583	0.440092	0.933794	0.814636	98.97161
52	−0.79538	−0.40033	−0.88197	0.97166	−0.22275	−0.28991	0.860131	0.825306	133.2404
53	−0.57416	0.665569	0.812562	0.150751	0.081135	0.857895	0.120364	0.111805	−119.48
54	0.62067	−0.12834	0.762917	0.828662	0.510371	0.517299	−0.5704	0.50425	53.34696
55	−0.76749	0.492631	−0.97983	0.296692	0.362353	0.711477	−0.30019	0.944011	−127.381
56	0.849134	0.314621	−0.22608	−0.77793	0.902875	−0.39672	−0.05598	0.839121	4.440011
57	0.386008	0.461785	0.279021	−0.3529	−0.89524	−0.39555	−0.99684	0.01459	47.2274
58	−0.90204	0.213319	−0.85119	−0.40301	−0.34397	0.682762	0.376978	0.431022	4.936711
59	0.24445	0.758876	−0.71051	0.310961	−0.36849	−0.62696	0.680268	0.775376	−51.891
60	−0.55873	0.226296	−0.4886	−0.33622	0.196823	−0.6504	0.088063	0.071867	14.9209
61	0.969918	0.017161	0.084454	0.241556	−0.64874	−0.96111	−0.58014	0.206049	−48.2017
62	0.915395	−0.58482	0.792565	−0.2735	−0.99951	−0.67445	−0.22295	0.994204	−1.50322

4.3. Comparison of the prediction capacity of GBSVRA and ELM-based models

Performances of the developed GBSVRA and ELM-SN model are presented in Figure 4 through mean absolute error (MAE) and root mean square error (RMSE) comparison during the two developmental stages of the models. The developed ELM-SN model shows overall best performance as inferred from its lowest values of RMSE, MAE and highest value of correlation coefficient (CC) during training and testing phases. ELM-SN model outperforms GBSVRA model on yardstick of RMSE [as shown in Figure 4 (a)] and on the basis of MAE [as presented in Figure 4 (b)] while training the models. ELM-SN model outperforms GBSVRA model during testing phase with performance superiority of 42.63%, 38.56% and 8.12%, respectively, using RMSE [as presented in Figure 4(c)], MAE [as shown in Figure 4(d)] and CC performance measure. Furthermore, ELM-SN model shows superior performance over ELM-SG model with 38.78% (using RMSE), 36.75% (using MAE) and 11.38% (using CC) improvement for testing set of data. The details of the performance of ELM-SN model over the two other developed models (GBSVRA and ELM-SG) are presented in Table 4.

Figure 4. Predictive capacity of developed models (a) using RMSE for training dataset (b) using MAE for training dataset (c) using RMSE for testing dataset (d) using MAE for testing dataset.

Table 4. Performances of the developed intelligent models

	dataset	CC	RMSE	MAE
GBSVRA	Training sample	1	0.001	0.001
	Testing sample	0.8697	97.6035	76.9928
ELM-SG	Training sample	0.895881	78.44218	52.26572
	Testing sample	0.838834	91.4621	74.78362
ELM-SN	Training sample	1	1.38E-12	1.15E-12
	Testing sample	0.946512	55.99322	47.3012
Superiority of ELM-SN over GBSVRA	Testing sample	8.115267	42.63196	38.56413
Superiority of ELM-SN over ELM-SG	Testing sample	11.37635	38.77987	36.74925

4.4. Predictions of the developed models and their comparison

Magnetic ordering temperatures for different classes of spinel ferrite magnetocaloric compounds obtained from the developed intelligent models are presented in Figure 5 for S1 = Zn_0.6Cu_0.4Fe₂O₄ (F. Hcini et al., 2021), S2 = Zn_0.4Ni_0.2Cu_0.4Fe2O₄ (F. Hcini et al., 2021), S3 = Zn_0.250Ni_0.250Mg_0.5Fe₂O₄ (F. Hcini et al., 2021), S4 = Ni_0.7Zn_0.3Fe₂O₄ (Bahhar et al., 2021), S5 = Ni_0.5Zn_0.5Fe2O₄ (Oumezzine et al., 2015), S6 = Ni_0.3Zn_0.7Fe₂O₄ (Oumezzine et al., 2015), and S7 = Cu_0.4Zn_0.6Fe2O₄ (Bahhar et al., 2021) compounds. The influence of nickel ions on MOT can be attributed to the preference of Zn²⁺ions for tetrahedral sites while both octahedral and tetrahedral sites contain the distribution of Fe³⁺ ions. Increase in the concentrations of Ni²⁺ ions in octahedral sites pushes Fe³⁺ ions from tetrahedral to octahedral sites and thereby increases tetrahedral-octahedral interactions which ultimately influence the magnetic ordering temperature (Oumezzine et al., 2015). The estimates of ELM-SN model captures the trend excellently followed by ELM-SG model while the developed GBSVRA model performs least.

Figure 5. Predictions of the developed intelligent models for different spinel ferrite magnetocaloric compounds [S1 = Zn_0.6Cu_0.4Fe₂O₄ (F. Hcini et al., 2021), S2 = Zn_0.4Ni_0.2Cu_0.4Fe2O₄ (F. Hcini et al., 2021), S3 = Zn_0.250Ni_0.250Mg_0.5Fe₂O₄ (F. Hcini et al., 2021), S4 = Ni_0.7Zn_0.3Fe₂O₄ (Bahhar et al., 2021), S5 = Ni_0.5Zn_0.5Fe2O₄ (Oumezzine et al., 2015), S6 = Ni_0.3Zn_0.7Fe₂O₄ (Oumezzine et al., 2015), and S7 = Cu_0.4Zn_0.6Fe2O₄ (Bahhar et al., 2021)].

The predictions obtained from the three developed models for S8 = Cu_0.2Zn_0.8Fe₂O₄ (Oumezzine et al., 2015), S9 = Ni_0.4Zn_0.6Fe₂O₄ (Oumezzine et al., 2015), S10 = Ni_0.35Zn_0.65Fe₂O₄ (Oumezzine et al., 2015), S11 = Ni_0.3Zn_0.7Fe₂O₄ (Oumezzine et al., 2015), S12 = Ni_0.25Zn_0.75Fe₂O₄ (Oumezzine et al., 2015), S13 = MnCeFeO₄ (Bahhar et al., 2021) and S14 = CoCeFeO₄ (Bahhar et al., 2021) spinel ferrite magnetocaloric compounds are presented in Figure 6. Change in the value of MOT as different magnetic cations are introduced into lattice structure of the parent spinel ferrite compounds for proper and adequate distributions within tetrahedral and octahedral sites of the compounds can be attributed to oxygen deficiencies and stoichiometry of the compound. Oxygen and defect vacancies are induced when Ce³⁺ are substituted in place of Fe³⁺ions in MnFe₂O₄ spinel ferrite compound (Bahhar et al., 2021). Hence, magnetic ordering temperature becomes modified due to the modification in the strength of indirect interactions caused by the defects. Figure 7 presents the estimated MOT for S15 = Ni_0.4Mg_0.3Cu_0.3Fe₂O₄, S16 = Cu_0.3Zn_0.7Fe₂O₄ (Bahhar et al., 2021), S17 = Mg_0.5Zn_0.5Fe₂O₄ (Fortas et al., 2020), S18 = Mg_0.3Zn_0.7Fe₂O₄ (Bouhbou et al., 2022),S19 = Mg_0.2Zn_0.8Fe₂O₄ (Bouhbou et al., 2022), S20 = Zn_0.7Ni_0.1Cu_0.2Fe₂O₄ (Bouhbou et al., 2022) and S21 = Mg_0.35Zn_0.65Fe₂O₄ (Bouhbou et al., 2022) compounds. For the magnesium substituted compounds, the difference in the ionic radius of Mg²⁺ and Zn²⁺ results in the observed variation in the value of MOT for the compounds. Furthermore, permutation of trivalent ions in divalent sites results into degradation of tetrahedral-octahedral super-exchange interactions (Bouhbou et al., 2022). The estimates obtained from ELM-SN model correlate well with the measured MOT due to the universal approximation strength of ELM algorithm coupled with the implemented Moore-Penrose generalized inverse method adopted by the algorithm.

Figure 6. Comparison of the estimates of the developed intelligent models for classes of spine ferrite magnetocaloric compounds [S8 = Cu_0.2Zn_0.8Fe₂O₄ (Oumezzine et al., 2015), S9 = Ni_0.4Zn_0.6Fe₂O₄ (Oumezzine et al., 2015), S10 = Ni_0.35Zn_0.65Fe₂O₄ (Oumezzine et al., 2015), S11 = Ni_0.3Zn_0.7Fe₂O₄ (Oumezzine et al., 2015), S12 = Ni_0.25Zn_0.75Fe₂O₄ (Oumezzine et al., 2015), S13 = MnCeFeO₄ (Bahhar et al., 2021), S14 = CoCeFeO₄ (Bahhar et al., 2021)].

Figure 7. Magnetic ordering temperature of spinel ferrite magnetocaloric compounds obtained from developed intelligent models [S15 = Ni_0.4Mg_0.3Cu_0.3Fe₂O₄, S16 = Cu_0.3Zn_0.7Fe₂O₄ (Bahhar et al., 2021), S17 = Mg_0.5Zn_0.5Fe₂O₄ (Fortas et al., 2020), S18 = Mg_0.3Zn_0.7Fe₂O₄ (Bouhbou et al., 2022),S19 = Mg_0.2Zn_0.8Fe₂O₄ (Bouhbou et al., 2022), S20 = Zn_0.7Ni_0.1Cu_0.2Fe₂O₄ (Bouhbou et al., 2022), S21 = Mg_0.35Zn_0.65Fe₂O₄ (Bouhbou et al., 2022)].

The influence of nickel, magnesium, cadmium and copper incorporation into tetrahedral and octahedral sites of spinel ferrite magnetocaloric compounds on MOT is shown in Figure 8 for S22 = Zn_0.375Ni_0.125Mg_0.5Fe₂O₄ (Fortas et al., 2020), S23 = Zn_0.250Ni_0.250Mg_0.5Fe₂O₄ (Fortas et al., 2020), S24 = Mg_0.6Cu_0.2Ni_0.2Fe₂O₄ (Fortas et al., 2020),S25 = Mg_0.6Cu_0.4Fe₂O₄ (Fortas et al., 2020), S26 = Ni_0.4Cd_0.3Zn_0.3Fe₂O₄ (Fortas et al., 2020),S27 = Ni_0.6Cd_0.2Zn_0.2Fe₂O₄ (Fortas et al., 2020), S28 = Zn_0.7Ni_0.3Fe₂O₄ (Oumezzine et al., 2015) and S29 = Zn_0.7Ni_0.2Cu_0.1Fe₂O₄ (Fortas et al., 2020) magnetocaloric compounds. The predictions of each of the developed three intelligent models show excellent correlation with the measured values.

Figure 8. Estimated MOT for different spinel ferrite magnetocaloric compounds [S22 = Zn_0.375Ni_0.125Mg_0.5Fe₂O₄ (Fortas et al., 2020), S23 = Zn_0.250Ni_0.250Mg_0.5Fe₂O₄ (Fortas et al., 2020), S24 = Mg_0.6Cu_0.2Ni_0.2Fe₂O₄ (Fortas et al., 2020),S25 = Mg_0.6Cu_0.4Fe₂O₄ (Fortas et al., 2020), S26 = Ni_0.4Cd_0.3Zn_0.3Fe₂O₄ (Fortas et al., 2020),S27 = Ni_0.6Cd_0.2Zn_0.2Fe₂O₄ (Fortas et al., 2020), S28 = Zn_0.7Ni_0.3Fe₂O₄ (Oumezzine et al., 2015), S29 = Zn_0.7Ni_0.2Cu_0.1Fe₂O₄ (Fortas et al., 2020)].

5. Conclusion

The magnetic ordering temperature (MOT) of spinel ferrite magnetocaloric compound is modeled from elemental compositions of the magnetic materials using hybrid genetic-based support vector regression algorithm (GBSVRA) and extreme learning machine (ELM). The applied magnetic field, ionic radii and concentrations of individual constituents serve as the descriptors to the models. Aside from the conventional divalent cations contained in spinel ferrite, the developed models can conveniently absorb two dopants distributed within tetrahedral and octahedral sites for MOT determination. The developed ELM-SN (with sine activation function) model outperforms GBSVRA and with performance superiority of 42.63% (using RMSE), 38.56% (using MAE) and 8.12% (using CC) on the testing set of data. The developed ELM-SN model also outperforms ELM-SG model with improvement of 38.78%, 36.35% and 11.38%, respectively, with RMSE, MAE and CC performance metrics. The estimates of ELM-SN model persistently agree with the reported experimental values of MOT as compared with the predictions of other models. The outstanding performance of the models developed would facilitate efficient and quick tuning of MOT of spinel ferrite for magnetic refrigeration and other closely related applications at a reduced cost and without experimental difficulties.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

section 3.2 contained the details of the raw data utilized for these findings with their references.

Additional information

Funding

This work was supported by the Ministry of Education – Kingdom of Saudi Arabia [IFP-A-2022-2-1-03.].