## 2 Method
### Fractional-Order Polynomial Prediction Model and Genetic Algorithm-Based Optimization
#### 2.2.2 Genetic Algorithm-Based Optimization  
In this paper, a gradient-free meta-heuristic genetic algorithm is preferred since the corresponding data is highly uncertain due to political crises and pandemic outbreaks. In the case of gradient-based optimization, unknown parameters can become easily unstable or can lose their robustness because of sharp changes. In addition, parameter constraints can be incorporated into the gradient-free optimization problem directly without solving a costate as in the gradient-based constrained optimization problems. Therefore, in this paper, a modified genetic algorithm is considered to search for the best solutions to the constrained optimization problem in Equation (5) by following the biological inspirations of the natural selection of the offspring. However, similar gradient-free meta-heuristic algorithms can be considered as well. However, since the genetic algorithm is well-known and easy to modify in the presence of multi-dimensional data and constraints, the genetic algorithm is preferred in this paper. Its detailed pseudo-code is presented with Algorithm 1. Since the optimization algorithm is a batch type, it can implicitly learn the overall character of the uncertainties, including the political crises and pandemic outbreaks. Therefore, the model presupposes that the uncertainty in the input data is averaged and smoothed. In addition, the optimization algorithm is constrained to avoid disproportional unknown model parameters. Henceforth, the model presupposes that the unknown parameters are inside the constrained parameter space. Moreover, since a closed-form solution is not available for such prediction problems, a higher-order polynomial basis is formed and validated with the training data. Thus, the model presupposes that the polynomial basis can also represent future data. ```
The parameter upper and lower limits \(\mathbf{w}_{\text{min}}^{\mathbf{T}_{r}}=-\eta^{r}/\max\left(\mathbf{b}^{\mathbf{T }_{r}}\right)\) and \(\mathbf{w}_{\text{max}}^{\mathbf{T}_{r}}=-\mathbf{w}_{\text{min}}^{\mathbf{T }_{r}}\). Initialized cost \(\mathbf{\hat{J}}^{\mathbf{T}}\) in Equation (5) and the best cost \(\mathbf{\hat{J}}^{\mathbf{T}_{r}}\). Initialized probability constant \(\beta\), crossover update rate \(\alpha\), mutation update rate \(\gamma\), mutation threshold \(\mu\). Initialization population length and search length. Output: Optimized unknown parameters \(\mathbf{w}^{\mathbf{T}_{r}}\in\mathbb{R}^{1\times N}\). Initialization: for\(i=1\) to population length \(1\)  Generate a random parameter population \(\mathbf{w}^{\mathbf{T}_{r}}\). \[\mathbf{w}^{\mathbf{T}_{r}}(i)=unifrnd\left(\mathbf{w}_{\text{max}}^{\mathbf{T }_{r}},\mathbf{w}_{\text{min}}^{\mathbf{T}_{r}},N\right)\] (6)  Construct the basis \(\mathbf{b}^{\mathbf{T}_{r}}(i)\) in Equation (1). Calculate the estimated output \(\mathbf{\hat{y}}^{\mathbf{T}_{r}}(i)\) in Equation (3). Assess the constrained current cost \(\mathbf{\hat{J}}^{\mathbf{T}_{r}}(i)\) in Equation (5). Compare the solution to the best cost \(\mathbf{\hat{J}}^{\mathbf{T}_{r}}\) as _if the current cost \(\mathbf{\hat{J}}^{\mathbf{T}_{r}}(i)\)_ is less than the best cost \(\mathbf{\hat{J}}^{\mathbf{T}_{r}}\)  Update \(\mathbf{w}^{\mathbf{T}_{r}}=\mathbf{w}^{\mathbf{T}_{r}}(i)\), \(\mathbf{\hat{J}}^{\mathbf{T}_{r}}=\mathbf{\hat{J}}^{\mathbf{T}_{r}}(i)\). end if Main Loop: for\(t=1\) to search length \(6\)  Determine the average cost \(\mathbf{\hat{J}}_{a}^{\mathbf{T}_{r}}=\mathbf{\hat{J}}^{\mathbf{T}_{r}}(t)/ \sum\limits_{l=1}^{t}\mathbf{\hat{J}}^{\mathbf{T}_{r}}(l)\). Calculate the probability as \(p(t)=e^{-\beta_{a}\mathbf{\hat{J}}^{\mathbf{T}_{r}}(t)}\). Perform the crossover. for\(k=1\) to population length\(/2\)  Determine the parent \(p^{1}\) with the Roulette Wheel selection: \[r=rand*\sum\limits_{l=1}^{t}p(l)\] (7) \[c(l)=c(l-1)+p(l)\quad\text{for $l=2$ to $t$}\] (8) \[p^{1}(k)=find(r\leq c)\] (9)  Repeat Equations (7)-(9) for the other parent \(p^{2}\).