Modeling the magnetocaloric effect of spinel ferrites for magnetic refrigeration technology using extreme learning machine and genetically hybridized support vector regression computational methods

Abstract Spinel ferrites are magnetic oxide materials with potentials to promote green technology in magnetic refrigeration which is known to be economically clean, energy saving and efficient. Maximum magnetic entropy change of spinel ferrites decides and controls the applicability as well as the strength of spinel ferrite magnetic oxide since it measures the hugeness of magnetocaloric effect. However, experimental determination of maximum magnetic entropy change requires intensive procedures, costly equipment and consumes appreciable time. Intelligent models are presented in this work using spinel-ferrite-molecular-based descriptors such as the ionic radii of spinel ferrites constituents, applied magnetic field and their concentrations. The developed intelligent models for prediction of spinel ferrite maximum magnetic entropy change include extreme learning machine (ELM) and hybrid genetic-algorithm-coupled support vector regression (GSVR). The developed ELM model has correlation coefficient (CC) and mean absolute error (MAE) of 98.45% and 0.117 J/kg/K, respectively, while the developed GSVR model has CC of 80.87% and MAE of 0.129 J/kg/J. The developed ELM model which is based on empirical risk minimization principle shows better performance over GSVR model that premises on structural minimization risk principle with improvement of 0.06%, 17.86% and 8.765% using root mean square error, CC and MAE yardsticks, respectively. Closeness of the estimates of the developed models with the experimental values is a strong indication of the potentials of the proposed intelligent methods in facilitating practical implementation of magnetic cooling refrigeration to solve energy crisis which promote efficiency and environmental friendliness.


Introduction
One of the viable alternative cooling technologies with future prospect to resolve global energy crisis and further replace the gas compression cooling system is the solid-state magnetic system of refrigeration due to its compactness, higher cooling efficiency and environmental friendliness (Makoed et al., 2019;Zhao et al., 2021).This cooling system utilizes magnetic materials having intrinsic characteristic known as magnetocaloric effect.Magnetocaloric effect describes and measures the entropy change that is induced by the alteration of the applied magnetic field (Neupane et al., 2022;Owolabi, 2023).Coupling between the spin system of magnetic material and the applied magnetic field ultimately induces magnetocaloric effect.Lack of magnetic material with the needed magnetocaloric effect hinders the applicability of magnetic refrigeration.Search for materials with huge magnetocaloric effect was previously centered on rare earth samples due to their high magnetic transition temperature as well as paramagnetic moment (Dong & Yin, 2020).Initially, gadolinium (lanthanide metal) was considered as refrigerant prototype with large known magnetocaloric effect and transition temperature of 293 K from paramagnetic to ferromagnetic; however, high cost of this material coupled with ease of its oxidation and difficulty in chemical synthesis constitute major setbacks for its wider application as refrigerant (Owolabi, 2023).The merits of spinel ferrites over this existing magnetic material include low cost of synthesis, low Eddy current loss and tunable magnetic moment transition temperature among others.Maximum magnetic entropy change (magnetocaloric effect) of spinel ferrites is modeled in this work from molecular properties descriptors which include ionic radii of elemental constituents, concentrations and applied magnetic field.
Spinel ferrites are potential alternatives to other existing magnetocaloric materials due to the ease of controlling their particle sizes and entropy change for attaining desired magnetic features (Felhi et al., 2018).Magnetocaloric effect of spinel ferrites can be controlled and tuned by altering the chemical compositions, distribution as well as the nature of cations present in octahedral and tetrahedral sites (Almessiere et al., 2019).Due to the possibility of altering physical as well as chemical properties of spinel ferrites through substitution of various cations in octahedral and tetrahedral sites of the compounds, spinel ferrites have various potential technological and industrial applications such as photocatalysis, drug delivery, cancer treatment, gas sensor and magnetic refrigeration among others (Alqahtani et al., 2022).Other ferrite-based compounds have demonstrated excellent magnetic and electrical properties (Ali & Singh, 2020;Hcini et al., 2018).However, spinel ferrite has demonstrated magnetic properties with enough potential in magnetic refrigeration technology.It is required to mention that other important parameters that render magnetocaloric materials useful for magnetic refrigeration aside from magnetocaloric effect include the magnetic ordering temperature and relative cooling power which has been treated elsewhere (Alqahtani et al., 2022).Spinel ferrites are classified as magnetic metal-oxide-based compounds with spinel structural formula AB 2 O 4 (Felhi et al., 2018).The crystallographic representation includes distribution of cations (metal) at tetrahedral and octahedral sites represented as A and B, respectively, and the presence of ferric in the chemical structure (Rajini & Ferdinand, 2022).Octahedral and tetrahedral sites coordination of oxygen atoms with metal cations constitutes structural description of spinel ferrite compounds.Physical and chemical properties of spinel ferrites strongly depend on the amount, nature and type of cations (metal) present in the crystallographic structure (Kefeni & Mamba, 2020).Spinel ferrite properties are tuned tuned when the magnetic moment ordering is influenced by the spin coupling existing between magnetic ions (3d shell electrons) and rare earth ions (4f shell electrons).The magnetic and physical properties of spinel ferrites can be enhanced through rare earth dopant inclusions.Rare earth ions dopants show preference for B site occupation in spinel crystallographic structure.This work employs extreme learning machine (ELM) and hybrid genetic-algorithm-based support vector regression (GSVR) to model maximum magnetic entropy change of spinel ferrites refrigerant.
Support vector regression (SVR) is a learning and pattern acquisition algorithm with structural risk minimization principle through efficient handling of epsilon insensitive loss function (Fan et al., 2021;Murillo-Escobar et al., 2019;Parsa & Naderpour, 2021).The algorithm employs quadratic optimization approach in attaining its global optimization and conveniently achieves better generalization ability even with small data samples (Oyeneyin et al., 2022).Excellent sparsity and nonconvergence to local solutions are among other unique characteristics of SVR algorithm.As a result of this uniqueness, SVR algorithm has enjoyed wider applicability in diverse areas of science, medical science and engineering (Dodangeh et al., 2020;Panahi et al., 2020;Suhaib et al., 2020).The predictive capacity of SVR algorithm depends heavily on hyper-parameter selection.Hence, the penalty factor, epsilon and kernel parameter are to be optimized for model accuracy enhancement (Owolabi et al., 2021;Owolabi & Abd Rahman, 2021;Shamsah & Owolabi, 2020).Different optimization algorithms such as particle optimization algorithm, gravitational search optimization algorithm and genetic optimization algorithm are among pool of optimization algorithms.Genetic optimization algorithm enjoys ease of global solution attainment and averts premature convergence and convergence to local solution (Adewumi et al., 2020;Olubosede et al., 2022).This work combines genetic optimization algorithm with SVR to develop hybrid model (GSVR) for predicting maximum magnetic entropy change of spinel ferrites for magnetic cooling application.
ELM is a supervised intelligent algorithm built with an architectural framework of single-layer networks (feed forward) (Alqahtani & Adewunmi, 2021;Owolabi & Gondal, 2018;Oyeneyin et al., 2021).Uniqueness of ELM as compared with traditional back propagation is its distinct capacity to randomly generate hidden layer biases as well as the input weights.As such, only model parameter that deserves tuning is the number of hidden nodes which tunes the network architecture appropriately (Christensen et al., 2022).Solution to the problem of regularized least squares conveniently computes the output weight vectors which join the output with hidden nodes.Hence, ELM-based model gains improved generalization strength in addition to a fast learning process.These distinct and excellent features of ELM have attracted wider applicability of the algorithm in addressing complex problems in science and engineering (Gao et al., 2022;Hu et al., 2020;Oyeneyin et al., 2021;Yan et al., 2022).This present work utilizes the uniqueness of ELM algorithm to model maximum magnetic entropy change of spinel ferrite magnetic materials for magnetic cooling technology using ionic radii, constituent concentration and applied magnetic field as model descriptors.
The remaining part of the manuscript is arranged as follows: section 2 formulates the mathematical background of the hybrid genetically based SVR algorithm and ELM.Section 3 presents and describes the data acquisition strategies and the employed computational methodology.Section 4 presents the results of each of the developed model, their comparisons and performance computation.Section 5 presents the conclusions.

SVR algorithm
SVR is an intelligent algorithm with statistical learning theory background and efficient capacity to acquire patterns using principle of minimization governed by structural risk mechanisms.Suitability of the algorithm to address regression tasks is controlled by the presence of binary classification in arbitrary properties space (Osuna et al., 1997).The algorithm attains generalized efficiency through minimization of distribution and training error after establishing non-linear relationship between the descriptors and target (Ren et al., 2020).Consider m number of spinel ferrite compounds whose maximum entropy change is to be determined through pattern acquisition training samples where χ 1 ðj ¼ 1 À mÞ represents the ionic radii and the concentration of the constituents that make up the spinel ferrite compounds, while ψ � 1 ðj ¼ 1 À mÞ is the measured maximum entropy change that controls the degree of magnetocaloric effect contained in the compounds for jth output and input vector.SVR algorithm approximates the acquired pattern through the function presented in equation ( 1) where α represents the bias term of the function, ψðχÞ is the estimated maximum entropy change using the regression model based on SVR, θðχÞ is the projection function which is not linear in nature and maps input descriptors χ to high feature space with characteristic high dimensionality and ω is the weight vector that is meant to be optimized and determined using SVR algorithm (Mohammadi & Mehdizadeh, 2020).The algorithm also determines the bias term using similar optimization strategies.In an attempt to minimize the arranged risk function, the algorithm minimizes the error function presented in equation (2) using the conditions and constraints embedded in equation (3).
where μ is the regularization factor which regulates the trade-off between model complexity and ensuring minimum error not beyond epsilon ε is maintained always for all training samples.The slack variables that further regulate the optimization function in case the threshold value of epsilon ε is to be exceeded are represented as β j and β � j .The input descriptors χ are mapped to feature space using θðχÞ.After dual optimization function transformation due to slack variables inclusion and convex optimization solution using LaGrange multipliers with mean Lagrangian coefficient $ j , the final regression function is expressed in equation (4) (Durgam et al., 2020).
Vector regression model typically invokes kernel function for its operation, while the fundamental kernel function employed in this present work is the Gaussian radial basis function expressed in equation ( 5) where χ j and χ i are input space vectors (Chibuike et al., 2022;Okoye et al., 2022).
where δ is the kernel parameter that controls the level of standard deviation contained in the Gaussian noise.

Genetic algorithm operational principle
Genetic algorithm is a population-based, global and stochastic optimization algorithm that mimics evolutionary principles and concepts introduced by Darwin (Holland, 1992).The algorithm generates individual in a random manner at the commencement of evolutionary processes with the production of offsprings that inherit the characteristic features of the parents for subsequent transition to the next generation (Ullah et al., 2021).The offspring's chance of surviving depends on the fitness of the parents since transformation from parents to the offspring involves features and qualities transfer between the concerned individuals.The mode of optimal global solutions obtained using genetic algorithm deviates from the conventional method of employing derivative information and thereby referred to as derivative-free method of optimization (Kumar et al., 2022).Hence, this optimization technique adequately locates global solutions in cases where the objective functions become noisy and non-smooth while the impossibility of extracting gradients from the models becomes apparent and finite differences approach of gradient estimation is laborious and time consuming.Execution of genetic algorithm involves vital steps which include initial population generation, fitness computation, selection operation, crossover operation, mutation operation and transition to the subsequent generation through population replacement (Alfarizi et al., 2022).Arbitrary solutions generated during initialization of population domain serve as the first step of the algorithm implementation.The generation and evolution of the population (known as chromosomes) follow the natural evolution principles proposed by Darwin (Liu et al., 2021;Tapia et al., 2021).The fitness function computes and weights the fitness of each of the chromosomes using defined objective function while transition to the subsequent generation is controlled through selection, crossover and mutation operations.Selection operator chooses parent chromosomes for reproduction while the crossover successfully generates descendent chromosomes.Inconsistencies present the pool of population is controlled through mutation operator (Motlagh et al., 2021).The processes continue until global convergence is attained and sustained for 50 consecutive iterations.

ELM operational principles
ELM is a learning algorithm that shares architectural resemblance with single-layer feed-forward learning networks (Hua et al., 2022;Liu et al., 2020).The distinct layers of the algorithm include the output, hidden and input layers (Bin Huang et al., 2006).Variables are fed into the learning architecture through the input layer, while the hidden layer interfaces the output and input layers and participates effectively in all the basic computations governing the operation principles of the algorithm.The target and desired parameters are housed in the hidden layer, while the outcomes of the computation are accessed from the output layer (Valipour & Shirgahi, 2022).Consider k samples of spinel ferrite compounds with α r ; ψ r ð Þ training set of samples in which α j represent the ionic radii as well as elemental compositions (descriptors), while ψ j stands for the maximum entropy change extracted from the literature for pattern generation and validation purposes.
The input descriptors are defined as α j 2 R m , while the measured maximum entropy change is defined as ψ j 2 R n .The number of neurons in input and output layers is defined as m and n respectively, while j ¼ 1; 2; . . .:; θ. θ represents the neurons in the hidden layer.The maximum entropy change ψ � computed through ELM algorithm is expressed in Equation ( 6) (Hua et al., 2022).
where δ j represents the jth hidden node output weight and defined as δ ¼ δ 1 ; . . .::; δ θ ½ � T θxn , while hðαÞ stands for hidden layer output vector with respect to input α and defined as hðαÞ ¼ h 1 ðαÞ; . . .:; h θ ðαÞ ½ �.The output vector contains a mapping feature with non-linear characteristics.Formal representation of hðαÞ is expressed in equation ( 7) where ς j 2 R d , τ j 2 R and f ðς j ; τ j ; αÞ is ELM activation function with characteristic hidden parameters (τ) where ς j is the input weight vector which is connected with the input layer, while τ j is the bias weight with j th node.In ELM, τ j and ς j are randomly selected at the commencement of the algorithm implementation.The compact form of equation governing ELM implementation is presented in equation ( 8) (Huang et al., 2020) where The hidden layer output matrix H is computed using relation presented in Equation ( 9), while the estimated maximum entropy change of spinel compounds is computed using the expression presented in equation ( 10) Equation ( 6) computes the weight matrix associated with the output layer through which estimates of the algorithm are computed (Valipour & Shirgahi, 2022).
where H � represents the Moore-Penrose generalized inverse of matrix H and can be effectively computed using H � ¼ H � HH T À � À 1 orthogonal approach.

Computational details and data acquisition
The computational details of the proposed hybrid GSVR are presented in this section.The details of ELM are also presented.This section further presents the details of the data acquisition.

Details of the data acquisition
Spinel ferrites compounds with different dopants incorporation were employed in this work for establishing relationships between MMEC, applied magnetic field, ionic radii and elemental concentrations.The utilized data for modeling and simulation were extracted from the literature (Bahhar et al., 2021;Bouhbou et al., 2017;Bouhbou et al., 2022;Felhi et al., 2018;Fortas et al., 2020;Hcini et al., 2021;Oumezzine et al., 2015;Zhao et al., 2021).The experimentally measured values of MMEC employed for validating the generalization as well as the prediction capacity of the developed models were extracted from the cited references.The employed data were culled from 31 spinel ferrites.In order to effectively implement the proposed models in this work for MMEC of spinel ferrite determination, equation ( 12) expresses a chemical structure that a spinel ferrite should assume prior to model implementation.Interestingly, the models developed in this work allow incorporation of two different dopants into tetrahedral sites of spinel ferrites which ultimately increase the property tuning change of the developed models.
where X represents the divalent metal in tetrahedral site, while Y and Z represent the dopants.The concentrations of the divalent metal as well as other dopants are represented as x, y and z, respectively.For example, if the developed models are to be implemented for determining MMEC of Ni 0.4 Mg 0.3 Cu 0.3 Fe 2 O 4 spinel ferrite compound, the descriptors to the models in this case include the ionic radius of nickel (which corresponds to X), ionic radius of magnesium (which corresponds to Y), ionic radius of copper (which corresponds to Z), elemental concentrations (0.4, 0.3 and 0.3 which correspond to x, y and z, respectively) and applied magnetic field.In a situation when one of the elements is missing or single dopant (instead of two) is desired to be incorporated in the parent spinel ferrite such as Mg 0.35 Zn 0.65 Fe 2 O 4 compound, the descriptors to the models in this case include the ionic radius of magnesium (which corresponds to X), ionic radius of zinc (which corresponds to Y), elemental concentrations (0.35 and 0.65 which correspond to x and y) and applied magnetic field while zero values are assigned to Z and z because of their absence in the chemical formula of the investigated spinel ferrite.Statistical analysis of the employed descriptors and experimental MMEC is presented in Table 1.
The presented standard deviation measures the consistency in the dataset as the data were extracted from different experimental measurements, while the contents of the employed dataset can be inferred from the presented mean values coupled with the maximum and minimum values.The presented coefficients of correlation measure the extent and degree of linear connections between the employed descriptors and the target.This establishes the need for non-linear modeling algorithm for connecting the descriptors with the measured MMEC.

Computational procedures for optimization of SVR parameters using genetic algorithm
Prediction precision and accuracy enhancements are keys to the effectiveness of any developed intelligent models with circumvention of the usual over-fitting and under-fitting problems.The parameters that tune the precision and accuracy of SVR-based model include the maximum error epsilon threshold, kernel parameter of any chosen kernel function and the regularization factor.Genetic evolutionary algorithm has been implemented within MATLAB computing environment for this parameter tuning due to its characteristic features such as quick global convergence and avoidance of premature convergence which settles to local solutions.Thirty-one available magnetocaloric spinel ferrite samples were subjected to randomization for even distribution of data-points prior to separation into training and testing sets in the ratio of 4:1, respectively.The choice of this partition ratio is due to few numbers of data-points characterizing this area of research work.The training samples help in pattern acquisition, while the future generalization capacity of the trained model was validated and assessed using testing magnetocaloric spinel ferrite samples.The detailed and stepwise procedures for computational strategies employed are summarized as follows: Step I: Population initialization within the search space: A set of probable solutions were generated within the search space for each of the hyper-parameters.The lower limits for the search space for the kernel parameter, regularization factor and epsilon error threshold were maintained at 0.1, 10 and 0.1, respectively, while the respective upper limits of the search space were set at 1.0, 500 and 0.5.The maximum attainable number of population was set as 100 after observing the sufficiency of the chosen maximum iteration number for global convergence attainment.Simulations were conducted for 20, 50, 100 and 200 population sizes, while the global convergence dependence of population size was investigated.
Step II: Fitness calculation through error function implementation: The chance of a chromosome to transit to other generation is determined by its fitness through objective function computation.Root mean square error (RMSE) between the measured maximum entropy change and the value estimated using SVR algorithm determines the chromosome's fitness.Therefore, each chromosome which houses the hyper-parameter (in the following orderliness: regularization factor, epsilon and kernel parameter) is fed into SVR algorithm after kernel function selection while the RMSE is computed.Fitness computational steps are as follow: (a) selection of kernel function among the available functions.Any selected kernel function must be symmetric, positive and semi-definite.These conditions are called Mercer's conditions.(b) One of the chromosomes from the population pool combines with the selected function in (a) and the training magnetocaloric spinel ferrite samples for training SVR algorithm.(c) Training RMSE was computed from the measured and estimated maximum entropy change (d) step (b) to (c) were repeated for other chromosomes until all the chromosomes in the population were explored (e) each of the trained algorithm in step (d) was validated by combining the support vectors acquired by each trained model with the descriptors in testing magnetocaloric samples.Therefore, testing-RMSE was computed for every trained model (f) rank the fitness of all the trained models in (e) using the principle that, the lower the value of testing-RMSE, the better the model.The details of the best algorithm were saved for population replacement and to ensure reproducibility of the models.
Step III: Population replacement through genetic algorithm operators: Old populations are replaced by new ones with better chance of global convergence.The operators that facilitate the replacement include the selection, crossover and mutation.The method of selection operation was stochastic for ensuring robust as well as quick global convergence.The crossover and mutation probabilities were set at 0.9 and 0.05, respectively, while the elitism was preserved.Crossover operator facilitates excellent characteristics transference from parent to offsprings through recombination and gene exchange.Mutation operator allows changing of genes while varieties within the population are enhanced by the operator.Incorporation of mutation operator circumvents and reduces the chances of local convergence.
Step IV: Stopping criteria: After repetitive algorithm procedures, the algorithm is subjected to termination conditions when global convergence has been achieved.The termination conditions include zero value of testing-RMSE, 50 consecutive values of testing-RMSE and maximum iteration attainment.The algorithm is brought to stoppage when any of the termination conditions is satisfied.The computational details are presented in Figure 1.

ELM computational details
Implementation of ELM algorithm for modeling MMEC of spinel ferrite magnetocaloric materials was conducted within MATLAB computing environment (MATLAB, 2015, MathWorks, Natick, MA, USA) using laptop personal computer (Intel(R) Core ™ i5-2450 M CPU @2.50 GHz 2.50 GHz).The descriptors to the models which include the externally applied magnetic field, ionic radius of divalent ions, ionic radii of two dopants and their respective concentration were randomized for even distribution of data-points and further divided into training and testing in the ratio of 7:3.Grid search approach was employed for selecting optimal value for number of neurons in a hidden layer for each of the chosen activation function since single parameter is to be optimized while implementation of heuristic algorithm becomes computational uneconomical (Karagoz, S., & Yildiz, 2017;Pholdee et al., 2017;Samala & Kotapuri, 2018;Yildiz & Yildiz, 2018).Stepwise procedures for computational implementation are described as follows: Step I: Initialization of number of hidden layer neurons with initiation of Mersenne Twister generator: The number (p) of neurons in hidden layer was initialized, while random numbers for the input weights and biases were also initiated through Mersenne Twister generator within MATLAB environment.
Step II: Optimum search for p for each of the available activation function: Number of neurons in a hidden layer was varied between 1 and 100 for each of the activation function which includes sigmoid, sine and triangular basis function using a grid search approach.In this optimization procedures, ELM algorithm was trained with training set of data and further validated with the assigned testing set of data.
Step III: Calculation of the element of hidden layer output matrix (H): Equation ( 9) computes the element of hidden layer output matrix for a range of hidden layer neurons and a chosen activation function coupled with training set of data.

Randomization of magnetocaloric materials and partitioning
Upper and lower limits of hyper-parameters Step IV: Calculation of weights connecting hidden layer with of output layer (δ): These weights are computed using equation ( 11) through Moore-Penrose generalized inverse matrix determination.

Testing-RMSE
Step V: Predictive strength of the model: Testing set of data combines with randomly generated input weights, biases and output weights to estimate MMEC of testing set of samples.The model performance is determined through testing set root mean square error (TSRMSE) determination for each of the number of neurons in hidden layer for every activation function.
Step VI: Stopping criteria: the algorithm stops when maximum number of hidden neuron is attained for each of the available activation function.The best model is selected with lowest TSRMSE while all the connecting weights and activation functions are saved.

Results and discussion
The outcomes of the developed intelligent models are presented in this section.Influence of the population size at different numbers of iteration on GSVR model is presented.Empirical relation based on ELM is also presented for subsequent practical implementation to new spinel ferrites.Performance comparisons between the models and model estimates are also presented.

Optimization influence of genetic algorithm on SVR parameters
The outcomes of the optimization of SVR parameters using genetic algorithm are presented in Figure 2. The parameters tuned included the regularization factor presented in Figure 2a, epsilon as presented in Figure 2b and kernel parameter presented in Figure 2c.For each of the optimized parameter, the chromosome population size was varied between 20 and 200 while 100 maximum iterations were maintained throughout the modeling phase.
The exploitation and exploration capacity of genetic algorithm was adequately explored to attain global solution with circumvention of possible premature or local convergence.Each of the population size results into similar convergence after little number of iterations which signifies robustness of the optimization algorithm.Error convergence is presented in Figure 2d for four different sizes of chromosomes exploring and exploiting the search space.Before 20 numbers of iterations, each of the chromosome size converges to a similar global convergence.Table 2 expresses the global value for each of the genetically optimized parameter.The optimum number of population was taken as 20 since it gives similar convergence with other investigated chromosome sizes.The Gaussian kernel function that helps in data transformation to feature space with characteristic high dimensionality is also presented in Table 2 with lambda value that controls the proper positioning of hyper-plane.

ELM connecting equation
Equation ( 13) presents the obtained ELM equation through which MMEC of spinel-ferrite-based compound can be computed.The weights required for implementing ELM equation are presented in Table 3.
where the estimated value of MMEC, sine activation function, output weight, input weights and biases are, respectively, represented as ψ � , f activation , δ, ς and τ.Several activation functions were explored and investigated while the optimum function is sine function.The optimum number of hidden nodes is 50, as presented in Table 3.The presented weights in Table 3 would definitely facilitate quick implementation of the developed ELM model for new set of spinel ferrites while determining their MMEC in cooling technology.

Performance comparison for the developed GSVR and ELM model
The performance of the developed GSVR and ELM model was assessed during training and testing phase, while the outcomes of the assessment during testing phase are presented in Figure 3. Figure 3a presents the assessment and comparison using RMSE yardstick, Figure 3b shows the comparison and assessment on the basis of coefficient of correlation (CC) while Figure 3c presents the comparison using mean absolute error (MAE) parameter.
For each of the performance parameter, the developed ELM model shows better performance as compared with GSVR model.Superiority in the performance of ELM model can be attributed to the intrinsic properties of the algorithm such as the implementation of empirical risk minimization as opposed to structural risk minimization principle adopted by SVR-based model, random generation of input and biases among others.Another interesting uniqueness of ELM model is the ease of implementation to newly synthesized spinel ferrites using the weight presented in Table 3 using Excel file or calculator while the implementation of SVRbased model needs initial retraining before implementation on newly synthesized spinel ferrites.

Estimates of the developed models as compared with the measured values
Estimated MMEC for a set of spinel ferrites are presented in Figure 4 using the developed GSVR and ELM models.Figure 4 further compares the estimates of each of the model with the measured samples.The measured values of all the investigated spinel ferrites are extracted from (Felhi et al., 2018;Fortas et al., 2020;Zhao et al., 2021)       Figure 6 presents the estimated MMEC for spinel ferrite compounds spanning using the developed intelligent models.The performance of ELM model further outperforms the developed hybrid GSVR model as can be observed from the closeness of the estimates.The outstanding performance of the developed ELM model can be attributed to the adopted empirical method of error minimization as well as random nature of chosen both input weights and biases.

Conclusion
ELM and hybrid GSVR models are developed and presented in this work for determining maximum magnetic entropy change of spinel ferrites for magnetic cooling technological application.The inputs to the models include the applied magnetic field, ionic radii of the spinel ferrites constituents and the elemental concentrations.The developed models allow incorporation of two different dopants in tetrahedral and octahedral sites while the descriptors are easily fetched which facilitate pre-laboratory simulation.The developed ELM model outperforms hybrid GSVR model during the training and testing stages on the basis of RMSE, MAE and CC.During testing phase, the computed values of RMSE, CC and MAE for the developed GSVR model are 0.1590 J/kg/J, 80.87% and 0.129 J/ kg/K, respectively, while the respective computed values for the performance parameters for ELM model are 0.1589 J/kg/J, 98.45% and 0.117 J/kg/K.Empirical expression governing the implementation of ELM model was obtained with weights for future implementation to newly synthesized spinel ferrites.The estimates of the developed ELM and GSVR model for various classes of spinel ferrites were compared with the measured values.The performance superiority of the developed ELM model over GSVR model can be attributed to the employed empirical risk principle of minimization and random choice of both input weights and biases.The outstanding precision demonstrated by the developed intelligent models would enhance and strengthen practical implementation of magnetic cooling technology since spinel ferrite with huge magnetocaloric effect can be identified using the developed models.
Figure 1.Computational methodology flow chart for GSVR model.

Figure
Figure 2. Convergence of GSVR model parameters as a function of iteration: (a) variation of regularization factor, (b) variation of epsilon over number of iteration, (c) variation of kernel parameter and (d) error variation at different numbers of population.
where ζ 1 toζ 7 are the input weights to each of the seven descriptors, τ is the biases and δ Figure 3. Testing model performance using (a) root mean square error, (b) correlation coefficient and (c) mean absolute error.

Fe 2 O
4 and Ni 0.4 Zn 0.6 Fe 2 O 4 that show very insignificant deviation.In the case of the developed GSVR model, only the estimated MMEC for Zn 0.6 Cu 0.4 Fe 2 O 4 , Ni 0.5 Zn 0.5 Fe 2 O 4 and Zn 0.4 Ni 0.2 Cu 0.4 Fe 2 O 4 spinel ferrite agreed excellently well with the measured values, while other spinel ferrites show slight deviation from the measured values.Another set of spinel ferrites compounds are presented in Figure 5 for comparison between the measured and the estimates of the two developed intelligent models.All the estimates of the developed ELM model agree excellently well with the measured value except Mg 0.3 Zn 0.7 Fe 2 O 4 that shows little deviation.For the developed GSVR model, Ni 0.3 Zn 0.7 Fe 2 O 4 and Mg 0.2 Zn 0.8 Fe 2 O 4 show excellent agreement, while other compounds show little deviation.

Table 3
Table 4 presents the performance parameters during training and testing phases.The developed GSVR model demonstrates better performance during training phase as compared with the testing phase of model development.Similar to the developed ELM model that even demonstrates 100% efficiency during the training phase.