Modeling the Specific Surface Area of Doped Spinel Ferrite Nanomaterials Using Hybrid Intelligent Computational Method

Spinel ferrites nanomaterials are magnetic semiconductors with excellent chemical, magnetic, electrical, and optical properties which have rendered the materials useful in many technological driven applications such as solar hydrogen production, data storage, magnetic sensing, converters, inductors, spintronics, and catalysts. The surface area of these nanomaterials contributes significantly to their targeted applications as well as the observed physical and chemical features. Experimental doping has shown a great potential in enhancing and tuning the specific surface area of spinel ferrite nanomaterials while the attributed experimental challenges call for viable theoretical model that can estimate the surface area of doped spinel ferrite nanomaterials with high degree of precision. This work develops stepwise regression (STWR) and hybrid genetic algorithm-based support vector regression (GBSVR) intelligent model for estimating specific surface area of doped spinel ferrite nanomaterials using lattice parameter and the size of nanoparticle as descriptors to the models. The developed hybrid GBSVR model performs better than STWR model with the performance improvement of 7.51% and 22.68%, respectively, using correlation coefficient and root mean square error as performance metrics when validated with experimentally measured specific surface area of doped spinel ferrite nanomaterials. The developed GBSVR model investigates the influence of nickel, yttrium, and lanthanum nanoparticles on the specific surface area of different classes of spinel ferrite nanomaterials, and the obtained results agree excellently well with the measured values. The accuracy and precision characterizing the developed model would be of immense importance in enhancing specific surface area of doped spinel ferrite nanomaterial prediction with circumvention of experimental stress coupled with reduced cost.


Introduction
Spinel ferrite nanomaterials have gained a significant and considerable attention lately due to their unique chemical, physical, magnetic, electrical, and optical features that are of great interest in many technological advancement and applications such as gas sensor, drug-delivery, photocatalysts, water splitting, spintronics, and supercapacitors [1][2][3][4]. Other important characteristics of spinel ferrite nano-materials that promote their wider applicability include stability, being less expensive, and ease of preparation [5]. The specific surface area of spinel ferrite nanomaterials contributes significantly to their technological applications especially during organic pollutant treatment [6][7][8]. The significance of specific surface area and other physical properties of catalysts to pollutant treatment has been treated elsewhere [9][10][11]. Tuning of specific surface area of spinel ferrite nanomaterials is often carried out experimentally through doping whereby foreign and external materials are incorporated into the parent spinel ferrite ceramic compounds and consequently leads to alteration in magnetic, electrical, and optical properties coupled with change in specific surface area of the nanomaterials [12][13][14][15]. This work models the specific surface area of spinel ferrite nanomaterials doped with foreign materials through stepwise regression-based model and hybrid genetic algorithmbased support vector regression (GBSVR) intelligent computational method using lattice parameter and the size of the nanomaterial as descriptors to the models.
Using crystal structure as a yardstick of classification, ferrite family can be hexagonal, garnets, or spinel ferrite [6]. The unique properties of spinel ferrites among other ferrite family distinguish them from others and have offered them a place in several technological applications. Interstitial sites in spinel ferrite over which cations are distributed include octahedral and tetrahedral sites [16][17][18]. Variation of charge distribution in these sites alters the specific surface area as well as other physical and chemical properties of the nanomaterials while the introduction of external materials in the crystal structure redistributes charges within the available sites [19,20]. Doping that accompanies specific surface area and other physical properties enhancement distorts the lattice structure of the parent spinel which makes lattice distortion becomes significant while modeling the influence of dopants incorporation on the specific surface area of the nanomaterials. Inclusion of the size of nanomaterial among the descriptors is of importance since the exhibited interesting physical and chemical properties of spinel ferrite nanomaterials are constrained to the nanoscale size of the materials and display distinct features when investigated outside nanoscale [7]. A hybrid of genetic and support vector regression algorithms presented in this work exclusively models the nonlinear relationship between the specific surface area and the descriptors which include distorted lattice parameter and the size of nanomaterial while the developed stepwise regression offers empirical relationship with significant deviation due to insufficient strength in addressing the nonlinear relationship existing between the descriptors and specific surface area. Support vector regression (SVR) belongs to the class of intelligent algorithms with fast computing potentials and excellent efficiency in addressing complex regression problems [21,22]. The algorithm is formulated using structural risk minimization principle and minimizes the empirical error through epsilon-insensitive loss function control [23][24][25]. Statistical learning theory upon which the algorithm forms the basis helps in error margin customization through hyperplane maximization. These features have made practical application of SVR algorithm in addressing real-life challenges and problems inevitable in various field of study [26][27][28][29]. The hyperparameters contained in SVR algorithm control the precision and accuracy of SVR-based algorithm and can be altered through various means which include grid search approach, manual search approach, or metal-heuristic approach [30,31]. Apart from robustness, precision, and avoidance of local minimal while using metal-heuristic hyperparameter optimization, time conser-vation is also a plus to metal-heuristic approach of hyperparameter selection. The metal-heuristic algorithm implemented in this work is genetic algorithm with characteristics of fast convergence and avoidance of premature convergence [32].
Stepwise regression (STWR) is a regression algorithm through which functions and equations that directly link the descriptors with the desired target are generated through either forward selection addition process or backward deletion procedures [33,34]. It weighs the significance of every descriptors based on some defined criteria before inclusion in the regression function. The algorithm also allows implementation of high-order polynomials as well as interactions of descriptors into regression function. The uniqueness of this algorithm has enjoyed a wide range of real life applications in science and engineering field [35][36][37][38][39]. This work also develops and implements STWR algorithm for modeling the specific surface area of doped spinel ferrite nanomaterials using lattice parameter and the size of nanomaterials as descriptors.
The road map for the remaining part of the manuscript is organized as follows: Section 2 presents mathematical formulation of support vector regression algorithm, stepwise regression algorithm, and physical principles governing the operation of population-based optimization genetic algorithm. Section 3 presents the computational hybridization of support vector regression and genetic algorithm as well as the descriptions with details of data acquisition. The results are presented and discussed in Section 4 of the manuscript with inclusion of the results of algorithm comparison, while Section 5 concludes and presents the summary of the main findings of the research work.

Mathematical Background of the Developed Algorithms
The mathematical formulation of the intelligent support vector regression algorithm is presented in this section coupled with the description of genetic algorithm. The section also contains the description of the implemented stepwise regression algorithm.
2.1. Support Vector Regression Intelligent Algorithm. Consider κ number of samples of data set defined as ðι j , S * j Þ, where j = 1, 2, ⋯, κ ⊂ ℝ d . The estimated specific surface area of doped spinel ferrite nanomaterials using the proposed support vector regression-(SVR-) based model is represented by S while the measured values of specific surface area from which patterns are to be acquired and drawn are represented by S * j . The descriptors to the proposed model which include the size of doped spinel ferrite nanomaterials and lattice distortion as measured by the lattice parameter are represented in regression equation as ι j . The support vector regression function is presented in Equation (1) [40,41].
Journal of Nanomaterials where S j ∈ ℝ d , ι j ∈ ℝ d , and ρis the bias of the regression function, and δ is the weight vector to be determined and optimized within support vector regression formulation. Euclidean norm kδk minimization in actualization of SVR objectives requires minimization of Equation (2) subjected to conditions and constraints itemized in Equation (3).
where C is the penalty factor that strongly influence the performance of the model.
Modification of complex optimization problem contained in Equation (2) includes nonzero positive slack variables (ψ j , ψ * j ) that have potentials of controlling further possible constraints that might hinder actualization of error minimization beyond the epsilon ε threshold [42,43]. Introduction of Lagrange multipliers (χ, χ * ) enhances dual problem transformation which aims at minimizing Equation (4) subjected to conditions presented in Equation (5).
χ and χ * are Lagrange multipliers. It should be noted that γðι j , ι i Þ in Equation (4) represents kernel function which can be polynomial, sigmoid, Gaussian, or other viable functions with distinct features. The function that best mapped the lattice distortion and size of nanoparticles to high feature space is Gaussian function presented in Equation (6) [44].
where ω is the kernel parameter.
The final regression function after transformation is presented in Equation (7).
The kernel parameter (ω) of the chosen kernel function, the maximum allowable deviation (ε) of the estimated specific surface area from the measured value, and the penalty factor are optimized using genetic algorithm in this research work.

Genetic Algorithm.
Genetic algorithm belongs to the class of well explored evolutionary optimization algorithm which mimics and replicates natural evolution [45,46]. Its development and implementation might be in binary or continuous form depending on the nature of the optimization problem to be handled since binary type caters for discrete space together with continuous space. Three operators that definitely feature in genetic algorithm principles include selection, crossover, and mutation. Before selecting best candidate through fitness function evaluation, a population matrix needs to be generated randomly and houses probable solutions within the search space [47,48]. Crossover operation produces offsprings from the defined probable solutions called parents after the implementation of selection operator. Mutation operation aids in genetic diversity maintenance and prevents local minimal convergence. The processes are repeated over a defined number of generations until stopping conditions are satisfied. Elitism inscribes robustness and effectiveness into genetic algorithm and enhances the optimality of the algorithm [49], although canonical genetic algorithm does not include elitism as an operator and among its functions is prevention of the best solution from undergoing mutation process. This helps in transference and preserving of best solutions from one generation to the subsequent generations.

Mathematical Background of Stepwise Regression.
Stepwise regression is a class of linear regression method which utilizes iterative construction of regression function in a step by step manner [33,34]. Stepwise regression equation is presented in matrix form as shown in Equation (8) in which X and β, respectively, represent descriptor vector and coefficient vector of X.
where y = ½y 1 , y 2 , It should be noted that the length of β is the length of descriptors which is two (the lattice parameter of doped spinel ferrite nanomaterials and the size of the nanomaterials) in this problem. Stepwise regression algorithm aims at iteratively and recursively determining the link between the descriptors and the target by weighing the contribution of each of the descriptors [50]. However, metrics adopted in adding and deleting descriptors include adjusted R 2 , Bayesian Information Criteria, Akaike Information Criteria, and Sum of Squared Error. The estimation strength of each of these criteria is judged using root mean square error (RMSE) of the tested samples. The procedures for adding and deleting descriptors are termed forward selection and backward elimination, respectively [33]. The choice of either of these methods does not in any way often affect the accuracy of the resulting regression function. However, one may save computational time than the other. In forward selection process, one descriptor is selected and added to regression 3 Journal of Nanomaterials function in the first step before adding the second descriptor after ensuring that the best fit is provided by its addition. The process continues until all the descriptors are added after satisfying some requirement such as adjusted R 2 , Bayesian Information Criteria, Akaike Information Criteria, and Sum of Squared Error. In backward elimination process, the development of regression function begins with inclusion of all the descriptors while one by one elimination follows after inability to meet up with the defined requirement and criteria. The choice of stepwise regression becomes significant in the addressed problem to ascertain the potential of nonlinear model in addressing and establishing the relationship between lattice parameter, nanomaterial size, and specific surface area of doped spinel ferrite nanomaterials.

Computational Methodology of the Developed Model
The details of the computational hybridization of support vector regression and genetic algorithm are presented here. This section also contains the details of the employed dataset and the results of the initial statistical analysis performed on the dataset.

Acquisition and Description of Spinel Ferrite
Nanomaterial Dataset. The lattice parameter of doped spinel ferrite nanomaterials obtained from X-ray diffraction analysis serves as the descriptor to the developed hybrid GBSVR model. The model also factors in the size of nanomaterial as its descriptor for estimating the specific surface area of spinel ferrite nanomaterial. The lattice distortion and the size of the nanomaterials with their corresponding specific surface area for forty spinel ferrite nanomaterials doped with different nanoparticles are extracted from the literature for the model development [5,7,8,18,[51][52][53][54][55]. The lattice strain as well as structural distortion in lattice structure of the parent spinel ferrite nanomaterial resulted from the introduced dopants definitely alters the material specific surface area. Table 1 presents the outcomes of statistical parameters evaluated on the dataset. Statistical parameters are evaluated on the lattice parameter, size of nanomaterial, and the measured specific surface area of doped spinel ferrite nanomaterials. The presented average (mean) values in the table as well as maximum and minimum values collectively reflect the overall content of the employed dataset. The significance of the standard deviation analysis presented in the table is to give a useful insight about the dispersion of the values of specific surface area and descriptors for different spinel ferrite nanomaterials doped with external materials. The correlation coefficients between the measured specific surface area and each of the descriptors reveal the potentialities of nonlinear model in establishing the relationship between specific surface area and descriptors. This gives an insight that the proposed hybrid genetic algorithm-based support vector regression (GBSVR) has potentials to outperform the proposed stepwise regression (STWR) that fails to incorporate nonlinearity in its operational modalities.

Computational Development of Proposed Hybrid Genetic
Algorithm-Based Support Vector Regression. Strong dependence of the performance of support vector regressionbased model on the combinatory choice of hyperparameters necessitates proper tuning and optimization of hyperparameters which include the maximum allowable deviation (epsilon), penalty factor, and kernel parameter of the best kernel function. The computational development of the proposed hybrid GBSVR model was conducted within computing environment of MATLAB. Dataset partitioning was carried out into training and testing set using 8 : 2 ratio while dataset randomization precedes data separation. Randomization promotes even, uniform, and just distribution of data points so that the possibility of validating and testing model on tasks that the model fails to learn due to insufficient data dispersion is prevented. With forty total number of doped spinel ferrite nanomaterials investigated, thirty-two doped compounds were employed in pattern acquisition during training stage, while the remaining eight doped compounds were utilized in evaluating the future performance of the model. The following procedures summarize the description of support vector regression hybridization with the genetic algorithm for performance enhancement.
Step 1. Population matrix: every chromosome in the population matrix carries information regarding the epsilon, penalty factor, and kernel parameter in a known and defined order. The kernel parameter is contained in the chosen kernel function and helps in smooth transformation of data to feature space where the construction of regression function takes place. The search space for the epsilon was set as 0.4 and 0.1 for upper and lower bound while setting outside this range resulted into perpetual running of the code. The kernel parameter ranges from 0.4 to 0.1 while the penalty factor was maintained between 1000 and 1. The right choice of hyperparameter range strengthens the exploitation and exploration capacities of the model and leads to appreciable time conservation.
Step 2. Chromosome fitness calculation: determination of chromosome fitness involves the implementation of SVR algorithm with training and testing set of data. Nonlinear

Journal of Nanomaterials
The chromosome with the lowest value of RMSE during testing stage is most fit, while other chromosomes were ranked in accordance to their fitness.
Step 3. Reproduction operation: reproduction involves selection of best chromosome with distinct qualities (usually characterized with lowest RMSE value) for transition to the next generation. In order to enhance fruitful and desired transition, a tournament selection operation with probability of 0.8 was implemented. Tournament selection involves random selection of few chromosomes from the whole population for tournament while the winners of the tournament transit to crossover stage of operation.
Step 4. Crossover stage: in crossover stage, new offsprings with desired qualities and potentials are formed from portions and subsequences of two parents. The probability value was set at 0.65 in order to achieve effective subsequences transfer from parent to the offsprings.
Step 5. Mutation stage: Altering positions that are randomly chosen within the chromosome is controlled through mutation operation, and the probability value was set at 0.009 in this present work.
Step 6. Stopping condition: the algorithm repeats the cycle between Step 1 and Step 5 until the same value of fitness is attained for consecutive sixty iterations.

Results and Discussion
The outcomes of the developed hybrid GBSVR model are discussed in this section. The performance of the developed GBSVR model is compared with that of STWR using different evaluation parameters. The sensitivity and convergence of the developed hybrid GBSVR model to the hyperparameter and the content of the population matrix is presented in this section. Comparison of the estimated specific surface area of spinel ferrite nanomaterials with the measured values for various dopants incorporation is also presented in this section.

Sensitivity and Convergence of the Developed Hybrid
GBSVR Model. The convergence of the developed hybrid GBSVR model at various iteration with different number of chromosomes in population matrix is presented in Figure 1.
The exploitation as well as exploration strength of the developed hybrid model is optimized as convergence is attained. It was observed from the convergence graph   Journal of Nanomaterials presented in Figure 1 that the developed model is robust and shows the same point of convergence for different number of chromosomes in the population matrix. The optimum population matrix was set at fifty from the obtained global solution convergence. The sensitivity of the developed GBSVR model to the penalty hyperparameter factor is presented in Figure 2 at various values of chromosomes in the population matrix. When ten numbers of chromosomes are navigating the search space, the penalty factor initializes at a value of 750 and increases steadily over ten iterations before begins the convergence. For fifty numbers of chromosomes, the chromosomes initialize at a value higher than the initial value of ten chromosomes, and similar convergence is attained. The convergence presented in Figure 2 shows that the global solution is well explored when there are many chromosomes within the search space. The sensitivity of the developed model with respect to the kernel parameter is presented in Figure 3. Similar trend is attained, and the model converged to global solution with 0.4 value of kernel parameter for Gaussian mapping function. The results of genetic optimization algorithm for each of the optimized hyperparameter are presented in Table 2.  The performance of the developed GBSVR and STWR models are compared during the training and testing phases of model development using correlation coefficient (CC) between measured and estimated specific surface area, mean absolute error (MAE), and root mean square error (RMSE). The outcomes of the comparison are presented in Figure 4.
The comparison on the basis of correlation coefficient on training dataset presented in Figure 4(a) shows that the developed hybrid GBSVR model displays enhanced performance over the developed STWR-based model with performance improvement of 56.25%, while the testing stage presented in Figure 4(b) shows performance superiority of 7.508%. Using RMSE for comparing the performance as    Figures 4(e) and 4(f). Table 3 contains the results of the performance measuring metrics for training and testing dataset. Correlation cross-plots between the measured specific surface area of doped spinel ferrite nanomaterials and the predicted values are presented in Figure 5 for training ( Figure 5(a)) and testing ( Figure 5(b)) set of data. The precision of the model can be judged from correlation cross-plots through alignment of data points. The results of the   spinel ferrite nanomaterial has been modeled using developed GBSVR, and the comparison between the measured and estimated specific surface area is presented in Figure 6. The effect of grain boundary deposition of lanthanum is initially observed from increase in specific surface area when the concentration of lanthanum nanoparticles is 0.06. Subsequently, after this concentration, the lanthanum nanoparticles get diffused into the parent crystal structure which leads to decrease in the value of specific surface area [5]. Few particles of iron (Fe 3+ ) in tetrahedral sites were replaced with lanthanum (La 3+ ) particles in octahedral sites, while this charge diffusion is responsible for the observed variation in specific surface area. The predicted specific surface area using the developed GBSVR model agree excellently well with the measured values [5].

Tailoring Specific Surface Area of Cd 1-x Y x Fe 2 O 4 Spinel
Ferrite Nanomaterial through Yttrium Dopants Using the Developed GBSVR Model. Figure 7 presents the effect of yttrium nanoparticles incorporation on specific surface area of doped Cd 1-x Y x Fe 2 O 4 spinel ferrite nanomaterial using the developed GBSVR model. Comparison of the measured specific surface area with the estimated values using the developed GBSVR model as presented in Figure 7 shows excellent agreement [52]. The observed variation of specific surface area due to yttrium incorporation can be inferred from the fact that substitution with higher ionic radius creates spaces in the configurations of spinel ferrite material. Cationic radius characterizing the nanoparticles substituted for yttrium surfers no alteration since the octahedral site of iron remains unchanged.

Importance of Nickel Nanoparticles in Enhancing Specific
Surface Area of Doped Co 1-x Ni x Fe 2 O 4 Spinel Ferrite Nanomaterial Using the Developed Hybrid GBSVR Model. Inclusion of nickel nanoparticles in the crystal structure of Co 1-x Ni x Fe 2 O 4 spinel ferrite nanomaterial further reveals the significance of cation distribution on the physical properties of spinel ferrite nanomaterials especially the specific surface area. As Fe 3+ ions is distributed among the available tetrahedral and octahedral sites, the incorporated nickel (Ni 2+ ) nanoparticle ions exist within tetrahedral sites while cobalt (Co 2+ ) ions reside in octahedral sites [8]. The comparison presented in Figure 8 shows that the experimental trend of nickel incorporation in lattice structure of Co 1-x Ni x Fe 2 O 4 spinel ferrite nanomaterial is well captured by the developed hybrid GBSVR model [8].

Conclusion
Specific surface area of doped spinel ferrite nanomaterials is modeled in this work using stepwise regression (STWR) algorithm and hybrid genetically based support vector regression (GBSVR) algorithm. The descriptors to the   Journal of Nanomaterials developed models include the lattice distortion due to dopant inclusion in lattice structure of spinel ferrite nanomaterials and the size of nanomaterials. The models were developed and validated using forty different spine ferrite nanomaterials subjected to incorporation of different dopants at various experimental conditions. The developed GBSVR model outperforms STWR model when evaluated using mean absolute error, correlation coefficient, and root mean square error metrics. The developed GBSVR model investigates the significance of lanthanum, yttrium, and nickel nanoparticle inclusions in different matrixes of spinel ferrite nanomaterials and the outcomes of the developed model agree excellently well with the measured specific surface areas. The outstanding performance demonstrated by the developed model is of enormous significance in tailoring the specific surface area of spinel ferrite nanomaterials to desired values needed for specific application with resources conservation and circumvention of experimental stress.

Data Availability
The raw data required to reproduce these findings are available in the cited references in Section 3.1 of the manuscript.

Conflicts of Interest
The authors declare that they have no conflicts of interest.