A Novel Hierarchical Extreme Machine-Learning-Based Approach for Linear Attenuation Coefficient Forecasting

. 2023 Jan 30;25(2):253. doi: 10.3390/e25020253

Algorithm 1 HELM Framework.

1:
Input: The matrix of the practice data X: ( $x_{i}$ )| $x_{i}$ ∈ Rd, i = 1, …, N;
2:
Function of a hidden node output: G( $a_{i}$ , $b_{i}$ , x);
3:
The number of hidden nodes: l;
4:
Size of iterations;
5:
Output: Hidden node matrix V: ( $a_{i}$ , $b_{i}$ ), i = 1, …, l; $β$ $_{i}$ ;
6:
Compute the H hidden layer output matrix;
7:
Calculate the matrix $H^{T} H$ ;
8:
Determine the constant $γ$ of the smooth convex function ∇, whose gradient relies on the largest eigenvalue;
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$y_{1} \overset{}{\leftarrow} β 0$ ∈ $R^{n}$ and $t_{1} \overset{}{\leftarrow} 1$ ;
10:
for (i = 0; i < x; i++) do
11:
Calculate $β$ $_{k}$ using Equations (12) and (13):
$β_{k} \overset{}{\leftarrow} p_{L} (y_{k}) \overset{}{\leftarrow} T_{α} (y_{k} - 2 \frac{1}{γ} H^{T} H + 2 \frac{1}{γ} H^{T} X)$ (12)

$T_{α} (x) \overset{}{\leftarrow} (| β | - α + s n g (β), α) \overset{}{\leftarrow} \frac{λ}{γ}$ (13)
12:
Calculate $t_{(k + 1)}$ using Equation (14):
$t_{(k + 1)} \overset{}{\leftarrow} (\frac{1 + \sqrt{1 + 4 t_{k}^{2}}}{2})$ (14)
13:
Calculate $y_{(k + 1)}$ using Equation (15)
$y_{(k + 1)} \overset{}{\leftarrow} β_{k} + (\frac{t_{k} - 1}{t_{k} + 1} (β_{k} - β_{(k - 1)}))$ (15)
14:
end for
15:
Return $β_{k}$