MetaHeuristic Optimization of LSTMBased Deep Network for Boosting the Prediction of Monkeypox Cases
Abstract
:1. Introduction
 A new approach is proposed based on optimized LSTM prediction to improve the accuracy of Monkeypox infection prediction.
 The proposed approach is compared with other ML models and optimization algorithms, and the results are recorded.
 The recorded results are analyzed using statistical methods such as Wilcoxon’s ranksum test and oneway analysis of variance to evaluate the statistical difference and significance of the proposed approach.
 The proposed approach can be generalized and tested for other datasets.
2. Related Works
3. The Proposed Methodology
Algorithm 1: The proposed prediction algorithm of Monkeypox confirmed cases. 

3.1. LSTM
3.2. AlBiruni Earth Radius Optimization Algorithm
Algorithm 2: BER optimization algorithm. 

3.2.1. Exploration Operation
 Moving towards the best solution : Using this strategy, the lone explorer in the group will look for promising new areas to explore in the immediate vicinity of where it now is. This is achieved by iteratively looking for a better choice (in terms of fitness) among the many possible alternatives in the immediate area. To do so, the BER study makes use of the following equations:$$\mathbf{r}=h\frac{cos\left(x\right)}{1cos\left(x\right)}$$$$\mathbf{D}={\mathbf{r}}_{1}(\mathbf{S}\left(t\right)1)$$$$\mathbf{S}(t+1)=\mathbf{S}\left(t\right)+\mathbf{D}(2{\mathbf{r}}_{2}1)$$
3.2.2. Exploitation Operation
 Moving towards the best solution: To move in the direction of the best solution, the following equation is employed.$$\mathbf{S}(t+1)={\mathbf{r}}^{2}(\mathbf{S}\left(t\right)+\mathbf{D})$$$$\mathbf{D}={\mathbf{r}}_{3}(\mathbf{L}\left(t\right)\mathbf{S}\left(t\right))$$
 Searching the area around the best solution: The area around the best answer is the most promising option (leader). This leads some people to look for improvements by exploring areas close to the optimal answer. The BER uses the following equation to carry out the aforementioned procedure.$${\mathbf{S}}^{\prime}(t+1)=\mathbf{r}({\mathbf{S}}^{*}\left(t\right)+\mathbf{k})$$$$\mathbf{k}=1+\frac{2\times {t}^{2}}{Ma{x}_{iter}^{2}}$$$$\mathbf{S}(t+1)=\mathbf{k}\ast {z}^{2}h\frac{cos\left(x\right)}{1cos\left(x\right)}$$
3.2.3. Selection of the Best Solution
4. Experimental Results
4.1. Dataset
4.2. Configuration Parameters
4.3. Optimization of Parameters in LSTM
4.4. Evaluation Criteria
4.5. The Achieved Results
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Algorithm  Parameter  Value 

AlBiruni Earth Radius (BER)  Iterations  500 
Number of runs  30  
Mutation probability  0.5  
Exploration percentage  70  
K (decreases from 2 to 0)  1  
Particle Swarm Optimization (PSO) [34]  Acceleration constants  [2, 12] 
Inertia ${W}_{max}$, ${W}_{min}$  [0.6, 0.9]  
Particles  10  
Iterations  80  
Grey Wolf Optimizer (GWO) [35]  a  2 to 0 
Iterations  80  
Wolves  10  
Genetic Algorithm (GA) [36]  Cross over  0.9 
Mutation ratio  0.1  
Selection mechanism  Roulette wheel  
Iterations  80  
Agents  10  
Whale Optimization Algorithm (WOA) [37]  r  [0, 1] 
Iterations  80  
Whales  10  
a  2 to 0 
Learning Rate  Hidden Nodes  Hidden Layers  

Lower bound  $1\times {10}^{5}$  1  1 
Upper bound  $1\times {10}^{1}$  20  10 
Optimized values  $3\times {10}^{2}$  7  2 
Metric  Value 

RMSE  $\sqrt{\frac{1}{N}{\sum}_{n=1}^{N}{(\widehat{{V}_{n}}{V}_{n})}^{2}}$ 
RRMSE  $\frac{RMSE}{{\sum}_{n=1}^{N}\widehat{{V}_{n}}}\times 100$ 
MAE  $\frac{1}{N}{\sum}_{n=1}^{N}\widehat{{V}_{n}}{V}_{n}$ 
NSE  $1\frac{{\sum}_{n=1}^{N}{({V}_{n}\widehat{{V}_{n}})}^{2}}{{\sum}_{n=1}^{N}{({V}_{n}\overline{\widehat{{V}_{n}}})}^{2}}$ 
MBE  $\frac{1}{N}{\sum}_{n=1}^{N}(\widehat{{V}_{n}}{V}_{n})$ 
R2  $1\frac{{\sum}_{n=1}^{N}{({V}_{n}\widehat{{V}_{n}})}^{2}}{{\sum}_{n=1}^{N}({\left(\right)}^{{\sum}_{n=1}^{N}}2}$ 
WI  $1\frac{{\sum}_{n=1}^{N}\widehat{{V}_{n}}{V}_{n}}{{\sum}_{n=1}^{N}{V}_{n}\overline{{V}_{n}}+\widehat{{V}_{n}}\overline{\widehat{{V}_{n}}}}$ 
r  $\frac{{\sum}_{n=1}^{N}(\widehat{{V}_{n}}\overline{\widehat{{V}_{n}}})({V}_{n}\overline{{V}_{n}})}{\sqrt{\left(\right)open="("\; close=")">{\sum}_{n=1}^{N}{(\widehat{{V}_{n}}\overline{\widehat{{V}_{n}}})}^{2}\left(\right)open="("\; close=")">{\sum}_{n=1}^{N}{({V}_{n}\overline{{V}_{n}})}^{2}}}$ 
Model  MSE  RMSE  MAE  R2  RRMSE  r  MBE  NSE 

BERLSTM (Proposed)  646.41  25.14  16.39  0.7  1.33  0.84  −3.75  0.65 
LSTM  655.33  27.31  18.6  0.59  1.66  0.833  3.79  0.59 
BILSTM  704.64  28.28  20.68  0.55  1.38  0.82  7.03  0.55 
GRU  643.15  27.08  17.62  0.61  1.33  0.83  1.79  0.61 
LSTMs  618.22  26.57  17.51  0.63  1.3  0.85  0.5  0.63 
BILSTMs  637.8  26.97  16.9  0.61  1.32  0.83  −0.65  0.61 
CONVLSTMs  728.28  28.73  17.41  0.53  1.41  0.8  0.72  0.53 
Model  MSE  RMSE  MAE  R2  RRMSE  r  MBE  NSE 

BERLSTM (Proposed)  480.53  20.82  15.25  0.73  1.36  0.83  0.06  0.61 
LSTM  586.06  26.09  19.24  0.45  1.486  0.78  7.67  0.45 
BILSTM  670.69  27.83  22  0.35  1.58  0.79  12.36  0.35 
GRU  519.42  24.7  17.51  0.53  1.41  0.81  6.16  0.53 
LSTMs  568.07  25.75  18.23  0.47  1.47  0.74  4.49  0.47 
BILSTMs  503.24  24.34  16.72  0.55  1.39  0.81  4.12  0.55 
CONVLSTMs  571.09  25.81  18.15  0.46  1.52  0.72  2.98  0.46 
BERLSTM  PSOLSTM  GWOLSTM  GALSTM  WOALSTM  

Num. values  8  8  8  8  8 
Range  0  1.9  1.8  2  2.3 
Maximum  20.82  22.8  23.1  23.9  24.2 
Minimum  20.82  20.9  21.3  21.9  21.9 
Mean  20.82  21.89  22.33  22.9  23.58 
Median  20.82  21.9  22.3  22.9  23.9 
Mean std. error  0  0.1797  0.179  0.189  0.2769 
Std. dev.  0  0.5083  0.5064  0.5345  0.7833 
25% Percentile  20.82  21.9  22.3  22.9  23.13 
75% Percentile  20.82  21.9  22.6  22.9  24.05 
Sum  166.6  175.1  178.6  183.2  188.6 
ANOVA Table  SS  DF  MS  F (DFn, DFd)  p Value 

Treatment (between columns)  34.77  4  8.694  F (4, 35) = 30.74  p < 0.0001 
Residual (within columns)  9.899  35  0.2828     
Total  44.67  39       
${\mathit{\mu}}_{\mathbf{BER}\text{}\mathbf{LSTM}}={\mathit{\mu}}_{\mathbf{PSO}\text{}\mathbf{LSTM}}$  ${\mathit{\mu}}_{\mathbf{BER}\text{}\mathbf{LSTM}}={\mathit{\mu}}_{\mathbf{GWO}\text{}\mathbf{LSTM}}$  ${\mathit{\mu}}_{\mathbf{BER}\text{}\mathbf{LSTM}}={\mathit{\mu}}_{\mathbf{GA}\text{}\mathbf{LSTM}}$  ${\mathit{\mu}}_{\mathbf{BER}\text{}\mathbf{LSTM}}={\mathit{\mu}}_{\mathbf{WOA}\text{}\mathbf{LSTM}}$  

p value (two tailed)  0.0078  0.0078  0.0078  0.0078 
Exact or estimate?  Exact  Exact  Exact  Exact 
Significant (alpha = 0.05)?  Yes  Yes  Yes  Yes 
PSOLSTM  GWOLSTM  GALSTM  WOALSTM  

Gaussian  Ambiguous  Ambiguous  Ambiguous  Ambiguous 
Bestfit values  
Amplitude  21.89  22.33  22.9  23.58 
Mean  20.82  20.82  20.82  20.82 
SD  2.465 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{32}$  2.465 $\times {10}^{32}$  2.465 $\times {10}^{32}$  2.465 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{32}$ 
95% CI (profile likelihood)  
Std  (Very wide)  (Very wide)  (Very wide)  (Very wide) 
Mean  (Very wide)  (Very wide)  (Very wide)  (Very wide) 
Goodness of Fit  
Degrees of Freedom  5  5  5  5 
R squared  0  0  0  0 
Sum of Squares  1.809  1.795  2  4.295 
Sy.x  0.6015  0.5992  0.6325  0.9268 
Constraints  
SD  SD > 0  SD > 0  SD > 0  SD > 0 
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Eid, M.M.; ElKenawy, E.S.M.; Khodadadi, N.; Mirjalili, S.; Khodadadi, E.; Abotaleb, M.; Alharbi, A.H.; Abdelhamid, A.A.; Ibrahim, A.; Amer, G.M.; et al. MetaHeuristic Optimization of LSTMBased Deep Network for Boosting the Prediction of Monkeypox Cases. Mathematics 2022, 10, 3845. https://doi.org/10.3390/math10203845
Eid MM, ElKenawy ESM, Khodadadi N, Mirjalili S, Khodadadi E, Abotaleb M, Alharbi AH, Abdelhamid AA, Ibrahim A, Amer GM, et al. MetaHeuristic Optimization of LSTMBased Deep Network for Boosting the Prediction of Monkeypox Cases. Mathematics. 2022; 10(20):3845. https://doi.org/10.3390/math10203845
Chicago/Turabian StyleEid, Marwa M., ElSayed M. ElKenawy, Nima Khodadadi, Seyedali Mirjalili, Ehsaneh Khodadadi, Mostafa Abotaleb, Amal H. Alharbi, Abdelaziz A. Abdelhamid, Abdelhameed Ibrahim, Ghada M. Amer, and et al. 2022. "MetaHeuristic Optimization of LSTMBased Deep Network for Boosting the Prediction of Monkeypox Cases" Mathematics 10, no. 20: 3845. https://doi.org/10.3390/math10203845