1. Introduction
The COVID19 pandemic has resulted in a huge worldwide catastrophe and has had a substantial impact on many lives across the world. The first instance of this deadly virus was reported in December 2019 from Wuhan, a Chinese province in [
1]. After emergence, the virus quickly became a worldwide epidemic, impacting many nations across the globe. Reverse transcriptionpolymerase chain reaction (RT PCR) is one of the most often utilized methods in the diagnosis of COVID19. However, since PCR has a diagnostic sensitivity of about 60–70%, radiological imaging techniques including computed tomography (CT) and Xray have been critical in the early detection of this disease [
2]. Therefore, the COVID19 diagnosis from CT and Xray images is an active and promising research domain, and additionally, there is much more space for improvements.
A few recent investigations have found alterations in Xray and CT imaging scans in individuals with COVID19 symptoms. For example, Zhao et al. [
3] discovered dilatation and consolidation, as well as groundglass opacities, in COVID19 patients. The fast increase in the number of positive COVID19 instances has heightened the necessity for researchers to use artificial intelligence (AI) alongside expert opinion to aid doctors in their work. Deep learning (DL) models have begun to gain traction in this respect. Due to a scarcity of radiologists in hospitals, AIbased diagnostic models may be useful in providing timely assistance to patients. Numerous research studies based on these approaches have been published in the literature; however, only the notable ones are mentioned here. Hemdan et al. [
4] suggested seven convolutional neural network (CNN) models, including enhanced VGG19 and Google MobileNet, to diagnose COVID19 from Xray pictures. Wang et al. [
5] classified COVID19 pictures from normal and viral pneumonia patients with an accuracy of 92.4%. Similarly, Ioannis et al. [
6] attained a class accuracy of 93.48% using 224 COVID19 pictures. The Opconet, an optimized CNN, was proposed in [
7] utilizing a total of 2700 pictures, giving an accuracy score of 92.8%. Apostolopoulous et al. [
8] created a MobileNet CNN model utilizing extricated features. In [
9], three different CNN models, namely inception v3, ResNet50, and InceptionResNet V2, were employed for classification. In [
10], a transfer learningbased method was utilized to classify COVID and nonCOVID chest Xray pictures utilizing three models such as ResNet18, ResNet50, SqueezeNet, and DenseNet121.
Although all of the abovementioned stateoftheart approaches use CNN, the methods do not take into consideration the spatial connections between picture pixels when training the models. As a result, when the pictures are rotated, certain resizing operations are performed, and data augmentation is executed owing to the availability of lower dataset sizes, the generated CNN models fail to properly distinguish COVID19 instances, viral pneumonia, and normal chest Xray scans. Although some degree of inaccuracy in recognizing viral pneumonia cases is acceptable, the misclassification of COVID19 patients as normal or viral pneumonia might confuse doctors, leading to failure of early COVID19 detection.
One of the promising ways for establishing an efficient COVID19 detection model based on DL is to generate a network with proper architecture for each COVID19 dataset. The no free lunch theorem (NFL) [
11], which claims that the universal method for tackling all realworld problems does not exist, proved as right in the DL domain [
12] and consequently standard DL model cannot render performance as good as models specifically tuned for COVID19 diagnosis. The challenge of finding appropriate CNN and DL structures for each particular task is known in the literature as CNN (DL) hyperparameters tuning (optimization), and a good way to do it is by using an automated approach guided by metaheuristics optimizers [
12,
13,
14,
15,
16,
17,
18,
19,
20]. The metaheuristicsdriven CNN tuning has also been successfully applied to COVID19 diagnostics [
21,
22,
23,
24].
However, the CNN tuning via metaheuristics is extremely time consuming because every function evaluation requires a generated network to be trained on large datasets for measuring solutions’ quality (fitness). Additionally, the CNN training process with standard algorithms, e.g., gradient descent (GD) [
25], conjugate gradienton (CG) [
26], Krylov subspace descent (KSD) [
27], etc., itself is very slow, and it can take hours to obtain feedback. Taking into account that the COVID19 diagnostics is critical and that the efficient network needs to be established in almost real time, more approaches for COVID19 early detection from Xray and CT images are required.
With the goal of shortening training time, while performing automated feature extraction, research presented in this manuscript adapts a sequential, twophase hybrid machine learning model for COVID19 detection from Xray images. In the first phase, a wellknown simple architecture alike LeNet5 CNN [
28] is used as the feature extraction to reduce structural complexities within images. The second phase uses extreme gradient boosting (XGBoost) for performing classification, where outputs from the flatten layer of the LeNet5 structure are used as XGBoost inputs. In other words, LeNet5 fully connected (FC) layers are replaced with XGBoost to perform almost realtime classification. The LeNet structure is trained only once, shortening execution time substantially more than in the case of CNN tuned approaches.
However, according to the NFL, the XGBoost, which efficiency depends on many hyperparameters, also needs to be tuned for specific problems. Consequently, this study also proposes metaheuristics to improve XGBoost performance for COVID19 Xray images classification. For the purpose of this study, modified arithmetic optimization algorithm (AOA) [
29], that represents a lowlevel hybrid between AOA and sine cosine algorithm (SCA) [
30], is developed and adapted for XGBoost optimization. The observed drawbacks of basic AOA are analyzed, and a method that outscores the original approach is developed. This particular metaheuristics is chosen because it shows great potential in solving varieties of realworld challenges [
31,
32]; however, since it relatively recenty emerged, it is still not investigated enough, and there are still many open spaces for its improvements.
The proposed twophases hybrid method for COVID19 Xray diagnosis is validated against the COVID19 radiography database set of images, which was retrieved from the Kaggle repository [
33,
34]. The classification is performed against three classes, namely normal, COVID19 and viral pneumonia. The viral pneumonia Xrays are also taken because only subtle differences with COVID19 Xray images exist. However, since the source of the COVID19 Xray diagnosis dataset is imbalanced toward the normal class and the aim of the proposed research is not oriented toward addressing imbalanced datasets, the COVID19 and viral pneumonia images are augmented, while the normal images are contracted from the original repository, and at the end each class, they contained 4000 observations.
The performance of the proposed methodology is compared with other standard DL methods as well as with XGBoost classifiers tuned with other wellknown metaheuristics. Additionally, the proposed modified AOA, before being adopted for XGBoost tuning for COVID19 classification, was first tested in optimizing challenging congress on evolutionary computation 2017 (CEC2017) benchmark instances.
Considering the above, this manuscript proposes a method that is guided by the two elemental problems for investigation:
The possibility of designing a method for efficient COVID19 diagnostics from Xray images based on the simple CNN and XGBoost classifier and
The possibility of further improving the original AOA apporach by performing lowlevel hybridization with SCA metaheuritiscs.
Established upon the experimental findings showed in
Section 4 and
Section 5, the contribution of the proposed study is fourfold:
A simple lightweight neural network has been generated that obtains a decent level of performance on the COVID19 dataset and executes fast;
An enhanced version of AOA metaheuristics has been developed that specifically targets the observed and known limitations and drawbacks of the basic AOA implementation;
It was shown that the proposed metaheuristics is efficient in solving global optimization tasks with combined, real and integer parameters types; and
The proposed COVID19 detection methodology from Xray images that employs the lightweight network, XGBoost and enhanced AOA obtains satisfying performance within a reasonable amount of computational time.
The sections of the manuscript are outlined as follows:
Section 2 provides a brief survey of the AI method employed in this study with a focus on CNN applications.
Section 3 explains the basic version of the AOA, points out its drawbacks and introduces the modified AOA implementation. Bound constrained simulations of the proposed algorithm on a challenging CEC2017 benchmark set are given in
Section 4. The experimental findings of the COVID19 early diagnostics from Xray images with the proposed methodology are provided in
Section 5, while the final remarks, proposed future work and conclusions are given in
Section 6.
3. Proposed Methodology
This section first shows a brief overview of original AOA metaheuristics, which is followed by its observed drawbacks and devised modified hybrid metaheuristics approach for the purpose of this study. Finally, this section concludes with a presentation of the twophase sequential DL and XGboost method, which is used for COVID19 Xray images categorization.
3.1. Arithmetic Optimization Algorithm
A novel method called arithmetic optimization algorithm (AOA) is a metaheuristic method which draw inspiration from mathematics fundamental operators introduced by Abuligah et al. [
29].
The optimization process of AOA initializes with
X, a randomly generated matrix, for which the single solution is represented as
${X}_{ij}$,
$1\le i\le N$, and
$1\le j\le n$, which represents the initial optimization space for solutions. The bestobtained solution is decided after each iteration and is considered a candidate for the best solution. The operations subtraction, addition, division, and multiplication control the computation of the nearoptimal solution areas. The search phase selection is calculated according to the Math Optimizer Accelerated (MOA) function applied during both phases:
where the
tth iteration function value is given as
$MOA\left(t\right)$, while the range is 1 to the maximum iterations number
T in which the current iteration is signified as
t.
$Min$ and
$Max$, respectively, represent the minimum and maximum accelerated function values.
The search space is randomly explored with the use of division (
D) and multiplication (
M) operators during the exploration phase. This mechanism is given with Equation (
8). When the condition
$r1>MOA$ is satisfied, the search is limited by the MOA for the current phase. The operator (
M) will not be applied until the first operator (
D) does not finish its task conditioned by
$r2<0.5$ as the first rule of Equation (
8). Otherwise, operator
D is substituted by the (
M) operator for the completion of the same task.
where the arbitrary small integer is
$\u03f5$, the fixed control parameter is
$\mu $, the
ith solution of the next iteration is
${X}_{i,j}(t+1)$, the current location
j of the current iteration’s
ith solution is
${X}_{i,j}\left(t\right)$, and the current best solution’s
jth position is
$best\left({X}_{j}\right)$. Standardly, the lower and upper boundaries of the
jth position are
$L{B}_{j}$ and
$U{B}_{j}$.
where the
tth iteration function value is denoted as the Math Optimizer Probability
$MOP\left(t\right)$, the current iteration is
t, the maximum iterations number is
T, and the fixed parameter is
$\alpha $ with the purpose of measuring the accuracy of exploitation over iterations.
The deep search of the search space for exploitation is afterwards performed by the search strategies employed with addition (
A) and subtraction (
S) operators. This process is provided in Equation (
10). The bounds of the first rule of Equation (
10) are
$r3<0.5$ which similarly links the operator (
A) to the operator (
S) as in the previous phase as (
M) to (
D). Furthermore, (
S) is substituted by (
A) to finish the task,
Conclusively, the nearoptimal solution candidates tend to diverge when $r1>MOA$, while they gravitate to nearoptimal solutions in case of $r1<MOA$. For the stimulation of exploration and exploitation, the values from $0.2$ to $0.9$ are incrementally increased for the $MOA$ parameter. Additionally, note that the computational complexity of AOA is $O(N\times (ML+1\left)\right)$ computational complexity.
3.2. Cons of Basic AOA and Introduced Modified Algorithm
The basic version of the AOA is regarded as a potent optimizer with a wide range of practical applications, but it stills suffer from several known drawbacks in its original implementation. These flaws are namely insufficient exploitation power and an inadequate intensity of exploration process. This is reflected in the fact that in some cases, AOA is susceptible to dwell in the proximity of the local optima and also to the slow converging speed [
32,
119,
120], as it can clearly be observed in CEC2017 simulations presented in
Section 4.
One of the root causes of these deficiencies is that the solutions’ update procedure in basic AOA is focused on the proximity of the single current global best solution. As discussed by [
119,
121], it results in an extremely selective search procedure, where other solutions depend on the solitary centralized guidance to update their position, with no guarantees to converge to the global optimum. Hence, it is necessary to improve the exploration capability of the basic AOA to escape the local optimums.
Due to the abovementioned cons, during the search process, the original AOA converges too fast toward the current best solution, and the population diversity is disturbed. Since the AOA’s efficiently depends to some extent on the generated pseudorandom numbers due to its stochastic nature, in some runs, when the current best individual in the initial population is close to optimum regions of the search domain, the AOA shows satisfying performance. However, when the algorithm is “unlucky” and the initial population is further away from optimum, the whole population quickly converges toward suboptimum regions, and the final results have lower quality.
Additionally, besides poor exploration, the AOA’s intensification process can be also improved. As already noted, the search is conducted mostly in the neighborhood of the current best individual, and exploitation around other solutions from the population is not emphasized enough.
The enhanced AOA proposed in this manuscript addresses both observed drawbacks by improving exploration, exploitation and its balance of the original version. For that reason, the proposed method introduces the search procedure from another metaheuristics and an additional control parameter that enhances exploration, but it also establishes better intensification–diversification tradeoff.
The authors were inspired by the lowlevel methodology of hybridization employing the principles from SCA to the AOA. This process results in satisfactory performance from both phases of the metheuristic solutions and a superior hybrid solution. The basic equations for position updating with the SCA are given (
11):
where the current option’s setting for the
ith measurement at the
tth model is
${X}_{i}^{t}$, arbitrary numbers
${r}_{1}$/
${r}_{2}$/
${r}_{3}$, the location factor placement in the
ith dimension is
${P}_{i}$, and the absolute value is given as
$\left\right$.
As stated above, after conducting extensive examination of the search equations of AOA and SCA algorithms, it was determined that AOA search equations are not sufficient for efficient exploitation, which to a large extent depends on the current best solution, and it is required to cover a wider search space. Hence, this research aimed to merge two algorithms combined with using a quasireflectionlearning based (QRL) procedure [
122] in the following way. Every solution lifecycle consists of two phases, where the solution performs an AOA search (phase one) and SCA search (phase two), which are controlled by the value of one additional control parameter.
Each solution is assigned a
$trial$ attribute, which is utilized to monitor the improvement of the solutions. In the beginning, after producing the initial population, all solutions start with an AOA search. In each iteration, if the solution was not improved, the
$trial$ parameter is increased by 1. When
$trial$ reaches the threshold value
$limit$ (control parameter in the proposed hybrid algorithm), that particular solution continues the search by switching to the SCA search mechanism. Again, every time when the solution is not improved,
$trial$ is increased by 1. If the
$trial$ reaches the
$2\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}limit$ value, that solution is removed from the population and replaced by the quasireflexiveopposite solution
${X}^{qr}$ of the solution
X, which is generated by applying Equation (
12) over each component
j of solution
X.
where
$\mathrm{rnd}\left(\right)open="("\; close=")">{\displaystyle \frac{LB+UB}{2}},X$ part of the equation has a role to generate a random value derived from the uniform distribution inside
$\left(\right)$, and
$LB$ and
$UB$ represent the lower and upper limits of the search space, respectively. This procedure is executed for each parameter of every solution
X within
D dimensions.
However, the replacement is not performed for the current best solution, because, practically, if the solution manages to maintain the best rank within $2\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}limit$ iterations, there is a great chance that this solution hits the right part of the search space. If such a replacement would have occured, then the search process might diverge from the optimum region.
It must be noted that when replacing the solution with its opposite, additional evaluation is not performed. The logic behind utilizing the quasireflexive opposite solutions is based on the fact that if the original solution did not improve for a long time, it was located far away from the optimum (or in one of the suboptimum domains), and there is a reasonable chance that the opposite solution will fall significantly closer to the optimum. Discarding socalled exhausted solutions from the population ensures stable exploration during the whole search process in the run. The novel solution starts its lifecycle as described above, with the $trial$ parameter reset to 0, and by conducting the AOA search first.
The value of the $trial$ threshold was determined empirically, and it is calculated by using the following expression: $limit=\frac{T}{2\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}N}$, where T denotes the maximal number of iterations, and N is the size of the population. Therefore, there is no need for the researcher to finetune this parameter.
For simplicity reasons, the introduced AOA method is named hybrid AOA (HAOA) and its pseudocode is provided in Algorithm 1. The introduced changes do not increase the complexity of the original AOA algorithm; hence, the complexity of the proposed HAOA is estimated as $O\left(N\right)=N+N\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}T$. Moreover, the HAOA introduces just one additional control parameter ($limit$), and it is automatically determined as it depends on T and N.
Algorithm 1: Hybrid arithmetic optimization algorithm. 
Initialize the parameters $\alpha $ and $\mu $. Initialize solutions’ positions randomly ($i=1,...,N$). Set $trial$ values of each solution to 0. Determine $limit$ value as $limit=\frac{T}{2N}$ while$t<T$do Compute the fitness function for the given solutions. Find the best solution so far. Update MOA and MOP values using Equations ( 7) and ( 9), respectively. for $i=1$ to $Solutions$ do if $trial<limit$ then Execute AOA search for $j=1$ to D do Generate a random number ($r1,r2,r3$) in interval [0, 1]. if $r1>MOA$ then Exploration phase if $r2>0.5$ then Apply the division operator (D, “÷”) Update the ith solutions’ positions using the first rule in Equation ( 8). else Apply the multiplication operator (M, “×”) Update the ith solutions’ positions using the second rule in Equation ( 8). end if else Exploitation phase if $r3>0.5$ then Apply the subtraction operator (S, “−”) Update the ith solutions’ positions using the first rule in Equation ( 10). else Apply the addition operator (A, “+”) Update the ith solutions’ positions using the second rule in Equation ( 10). end if end if end for Compare the old solution and updated solution and increment $trial$ if needed. else if $trial<2\ast limit$ then Execute SCA search for $j=1$ to D do Update positions according to Equation ( 11). end for Compare old solution and updated solution and increment $trial$ if needed. else if i is not the current best solution then Remove solution ${X}_{i}$ from the population. Replace ${X}_{i}$ with quasireflexiveopposite solution ${X}_{i}^{qr}$ produced with Equation ( 12). Reset $trial$ parameter to value 0. end if end if end for $t=t+1$ end while Return the best solution.

3.3. Deep Learning Approach for Image Classification
As is it was already mentioned in
Section 1, the proposed approach is executed in two phases, where the first phase performs feature extraction and the second phases employs XGBoost for performing classification.
In the first phase of the proposed approach, a simple CNN architecture, similar to LeNet5 [
28] that consists of 3 convolutional and 3 max pooling layers, followed by 3 fullyconnected layers, is employed. This network structure was determined empirically with the goal of being as simple as possible (allowing easier training and fast execution), while achieving a decent level of performance on the COVID19 dataset, by performing hyperparameters optimization during the preresearch phase via a simple grid search. The hyperparameters that were tuned included the number of convolutional layers (range
$[2,5]$, integer), number of cells in convolutional layers (range
$[3,36]$, integer), number of fully connected layers (range
$[2,5]$, integer) and learning rate (range
$[0.00001,0.1]$, continuous). The determined network structure is as follows: the first convolutional layer uses 32 filters with 3 × 3 kernel size, while the second and third convolutional layers employ 16 filters with 3 × 3 kernels, which is followed by 3 dense layers. The complete CNN network structure is shown in
Figure 2.
All images are resized to
$32\times 32$ pixel size and used as CNN input, where the input size is
$32\times 32\times 3$. The convolutional layers’ weights are pretrained on a COVID19 dataset, as described in
Section 5.1 with the Adam optimizer and a learning rate (
$\eta $) of 0.001,
sparseCatagoricalCrossEntropy loss function and a batch size of 32 over 100 epochs. The CNN uses a training set and validation set, which is a 10% fraction of the training data, and an early stopping condition with respect to validation loss with patience set to 10 epochs.
Due to the stochastic nature of the Adam optimizer, the whole training process is repeated 50 times, and the best performing pretraining model is used for the second phase. Training and validation loss for the best model during the training is shown in
Figure 3, where it can be seen that the due to early stopping criteria, training terminated after only 60 epochs.
After determining the suboptimal weights and biases of the used simple CNN in the first phase, in the second phase, all fully connected layers from the CNN are removed, and the outputs from CNN’s flatten layer are used as inputs for the XGBoost classifier. Therefore, all CNN’s fully connected layers are replaced with XGBoost, where XGBoost inputs represent features extracted by the convolutional and maxpooling layers of CNN.
However, as it was also pointed out in
Section 1, the XGBoost should be optimized for every particular dataset. Therefore, the proposed HAOA is used for XGBoost tuning, where each HAOA solution is of length 6 (
$L=6$), with every solution’s component representing one the XGBoost hyperparameters.
The collection of XGBoost hyperparameters that were addressed and tuned in this research is provided below, together with their boundaries and variable types:
Learning rate ($\eta $), limits: $[0.1,0.9]$, category: continuous;
$Min\_child\_weight$, limits: $[0,10]$, category: continuous;
Subsample, limits: $[0.01,1]$, category: continuous;
Collsample_bytree, limits: $[0.01,1]$, category: continuous;
Max_depth, limits: $[3,10]$, category: integer; and
$Gamma$, limits: $[0,0.5]$, category: continuous.
The parameter count required by softprob objective function (‘num_class’:self.no_classes) is further being passed as the parameter to XGBoost as well. All other parameters are determined and set to default XGBoost values.
Finally, the hybrid proposed approach is named after the used models—CNNXGBoostHAOA, and its flowchart is depicted in
Figure 4.
4. CEC2017 BoundConstrained Experiments
The XGBoost tuning belongs to the group of NPhard global optimization problems with mixed, real values and integer parameters (see
Section 3.3). However, to prove the robustness of the optimizer, it should be first tested on a larger set of global optimization benchmark instances before being validated against the practical problem such as XGBoost hyperparameters optimization.
Therefore, the HAOA was validated on exceedingly challenging global optimization benchmark functions from the CEC2017 testing suite [
123] with 30 parameters. The total number of instances is 30, and they are divided into 4 groups: from
$F1$ to
$F3$—unimodal, from
$F4$ to
$F10$—multimodal, hybrid functions are instances from
$F11$ to
$F20$, and finally, the most challenging functions are the composite ones that include instances from
$F21$ to
$F30$. The composite benchmarks exhibit all characteristics of the previous 3 groups; plus, they have been rotated and shifted.
The
$F2$ instance was discarded from experimentation due to its unstable behavior, as pointed out in [
124]. The full specification of benchmark functions including name, class, parameters search range and global optimum value are shown in
Table 1. More details, such as its visual representation, can be seen in [
123].
All simulations were performed with 30dimensional CEC2017 instances (
$Dim=30$), and results for obtained mean (average) and standard deviation (std) averaged over 50 separate runs are reported. These two metrics are the most representative due to the stochastic behavior of metaheuristics. A relatively extensive evaluation of metaheuristics performance for the CEC2017 benchmark suite is provided in [
125], where stateoftheart improved harris hawks optimization (IHHO) was introduced; therefore, a similar experimental setup as in [
125] was used in this study.
The research proposed in [
125] validated all approaches in simulations with 30 individuals in the population (
$N=30$) and 500 iterations (
$T=500$) throughout one runtime. However, some metaheuristics spare more
FFEs in one run, and setting the termination condition in terms of iterations may not be the most objective strategy. Therefore, to compare the proposed HAOA with other methods without biases, and at the same time to be consistent with the abovementioned study, this research uses 15,030
FFEs (
$N+N\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}T$) as the termination condition.
Additionally, most of the methods presented for validation purposes in [
125] were also implemented in this study with the same adjustments of control parameters. The comparison between the proposed HAOA and the following methods was performed: basic AOA, SCA, cuttingedge IHHO [
125], HHO [
126], differential evolution (DE) [
127], grasshopper optimization algorithm (GOA) [
128], gray wolf optimization (GWO) [
129], moth flame optimization (MFO) [
130], multiverse optimizer (MVO) [
131], particle swarm optimization (PSO) [
83] and whale optimization algorithm (WOA) [
132].
Results for the CEC2017 simulations are displayed in
Table 2. The text in bold emphasizes the best results for every performance indicator and instance. In the case of equal performance, these results are also bolded. Regardless whether the experimentation in [
133] was performed with
T as the termination condition, the results reported in this study are similar. However, due to the stohastic behavior of the optimizer, subtle differences exist.
The best mean results for 21 functions were achieved by the HAOA, and they include
$F1$,
$F3$,
$F5$,
$F6$,
$F7$,
$F8$,
$F11$,
$F12$,
$F13$,
$F15$,
$F17$,
$F19$,
$F20$,
$F21$,
$F22$,
$F23$,
$F25$,
$F26$,
$F28$,
$F29$, and
$F30$. The functions are shown in
Table 2. The second best approach proved the best cuttingedge IHHO, and in some tests, the IHHO showed better performance than HAOA, while in others, the results of HAOA and IHHO were tied. The HAOA and IHHO obtained the same mean indicator values in the following tests:
$F3$,
$F6$,
$F19$,
$F21$, and
$F29$. The small number of cases in which the HAOA performed worse than the IHHO includes
$F4$ and
$F14$ experiments. There are also some cases where other methods achieved the best results, e.g., the
$F9$ instance, where MVO and PSO showed superior performance. Lastly, the HAOA tied DE in the cases of
$F13$ and
$F15$ instances.
Additionally, it is very important to observe that the original AOA never beat HAOA. Moreover, there are instances where the HAOA tremendously outscored AOA, even by more than 1000 times, e.g., in the function $F1$ test. Finally, it is also significant to compare HAOA and SCA, because the HAOA uses SCA search expressions. In all simulations, the HAOA outperformed SCA for both indicators. Accordingly, it can be concluded that the HAOA successfully managed to combine the advantages of basic AOA and SCA methods as a lowlevel hybrid approach.
The magnitude of results’ variances between the HAOA and every other method implemented in CEC2017 simulations can be determined from a Friedman test [
134,
135] and twoway ranks variance analysis. This was performed for the reasons of statistical importance of an improvement’s proof that is more thorough than simply putting outcomes into comparison.
Table 3 summarizes the results of the Friedman test over 29 CEC2017 instances for 12 compared methods.
Observing
Table 3, the HAOA undoubtedly performs better than any of the other 11 algorithms taken into account for comparative analysis. As expected, the second best approach is IHHO, while the original AOA and SCA take the ranks of 6 and 11, respectively. Additionally, the calculated Friedman statistics
${\chi}_{r}^{2}$ is
$21.672$, and as such, it is greater than the
${\chi}^{2}$ critical value with 11 degrees of freedom (
$1.9675\times {10}^{1}$) at the threshold level of
$\alpha =0.05$. The conclusion of this analysis is that the null hypothesis (
$H0$) can be rejected, implying that the HAOA achieved results which are substantially better than other algorithms.
The convergence speed visual difference between the proposed HAOA and AOA, SCA, as well as between the other three bestperforming metaheuristics, IHHO, DE and PSO for
$F4$,
$F6$,
$F11$,
$F17$,
$F22$ and
$F28$ instances, is shown in
Figure 5. From the sample functions convergence graphs, it can be observed that the HAOA converges on average faster than other methods, which is particularly emphasized in cases of
$F4$,
$F6$ and
$F11$ instance. It can also be seen that the results’ quality generated by HAOA is much higher than its base algorithms, AOA and SCA.
6. Conclusions
Fast diagnostics is crucial in modern medicine. The ongoing COVID19 epidemic has shown how important it is to quickly determine whether or not a patient has been infected, and fast treatment is often the key factor to saving lives. This paper introduces a novel early diagnostics method to detect the disease from lungs Xray images. The proposed model utilizes a novel HAOA metaheuristics algorithm, which was created by hybridizing AOA and SCA algorithms with a goal to overcome the deficiencies of the basic variants. The solutions in the proposed hybrid algorithm start by performing an AOA search procedure, and if the solution does not improve over the iterations, it will switch to the SCA search mechanism (controlled by the additional $trial$ parameter). If the solution still does not improve, ultimately, it will be replaced by a quasireflective opposite solution, as defined by the QRL procedure.
The HAOA algorithm was put to test on a set of hard CEC2017 benchmark functions and compared to the results of the basic AOA and SCA and another cuttingedge metaheuristics algorithm. It can be concluded that the HAOA undoubtedly achieves a higher level of performance than the other eleven tested algorithms. After proving the superior performance on the benchmark functions, the algorithm was employed in the machine learning framework, consisting of the simple CNN used for feature extraction and an XGBoost classifier, where HAOA was used to tune the XGBoost hyperparameters. The model was named CNN–XGBoost–HAOA, tested on a large COVID19 Xray images benchmark dataset, and compared to eight other metaheuristics algorithms used to evolve the XGBoost structure. The proposed CNN–XGBoost–HAOA obtained predominant accuracy of almost 99.4% on this dataset, leaving behind all other observed models.
The contribution of the proposed research can be defined on three levels. First—a simple lightweight network was generated, that is easy to train, operates fast and achieves decent performance on the COVID19 dataset, where the XGBoost classifier was used instead of fully connected layers. Second—AOA metaheuristics was improved and used in the model. Finally, the whole model has been adapted to the COVID19 dataset. The limitations of the proposed work are closely bound to these three levels of contributions. First, it was possible to execute more detailed experiments with the hyperparameters of the simple neural network to begin with, and it was also possible obtain another light structure that could have an even better level of performance; however, this was out of the scope of this work. Second, each metaheuristics algorithm can be modified in an infinite number of theoretically possible improvements (minor modifications and/or hybridization), leading to the conclusion that in theory, the level of improvements of the basic AOA could be even higher without increasing the complexity of the algorithm. It was also possible to include other XGBoost parameters to the tuning process, as there are many of them, but it was not possible to cover all this with just one study. Finally, experiments were executed with just one dataset, which has been balanced. The experiments with imbalanced datasets were not executed, because addressing imbalanced datasets was not goal of presented study.
Based on these encouraging results, the future work will be centered around gaining even more confidence in the suggested model by testing it further on the additional reallife COVID19 Xray datasets before considering the practical implementation as a part of the system that could be used in the hospitals to help in early COVID19 diagnostics.