Algorithm 1. PSO-RBFN-SA

Offline phase

$n_{spectrum}$ prepared spectra are used as samples for RBFN training.

For $\alpha =1,2,\dots ,n_{spectrum}$

PSO is performed to the spectrum $F_{(\alpha )}$, and the population of particles is set as $n_{particle}$.

For $l=1,2,\dots ,n_{particle}$

$P_l$ represents the current solution, which is initialized as a random solution in the solution space: $$P_l(1)=[a_0,a_1,\dots ,a_{\tau },w_1,w_2,\dots w_m]$$ (14)

$V_l$ represents the current velocity, which is initialized as a random velocity: $$V_l(1)=[v_1,v_2,\dots ,v_{\tau +m}]$$ (15)

$P_l^s$ represents the best solution it has achieved so far, which is initialized as: $$P_l^s(1)=P_l(1)$$ (16)

End

The maximum iteration of PSO is set as $n_{PSO}$.

For $t=1,2,\dots ,n_{PSO}$

The global best solution $P^g$ is defined as: $$z(P^g(t))=\text{MIN}(z(P_l^s(t)))l=1,2,\dots ,n_{particle}$$ (17)

For $l=1,2,\dots ,n_{particle}$

The weighted particle velocity is updated as: $$V_l(t+1)=\eta (t)V_l(t)+c_1{R}_1[P_l^s(t)-P_l(t)]+c_2{R}_2[P^g(t)-P_l(t)]$$ (18) where $R_1$ and $R_2$ are two separate random number between 0 and 1, while $c_1$ and $c_2$ are acceleration constants, representing the weight of acceleration terms that pull each particle toward the local best solution and the global best solution. $\eta (t)$ is the inertia weight for balancing the global and local influences, which is defined as: $$\eta (t)=0.9-\frac{t}{n_{PSO}}\times 0.5$$ (19) where $\eta (t)$ linearly decreases through the course of the run. A large inertia weight facilitates a global search while a small inertia weight facilitates a local search. Accordingly, the optimization process can converge to the neighborhood of global optimal solution smoothly at the prophase and converge to the global optimal solution quickly at the anaphase.

The solution of each particle is updated as: $$P_l(t+1)=P_l(t)+V_l(t+1)$$ (20)

The best position of particle is calculated as: $$P_l^s(t+1)=\{\begin{array}{cc}{P}_l^s(t)& z(P_l(t+1))\ge z(P_l^s(t))\\ P_l(t+1)& z(P_l(t+1))<z(P_l^s(t))\end{array}$$ (21)

End

End

The global optimization result of $F_{(\alpha )}$ is recorded as $P_{(\alpha )}^g$.

End

RBFN is trained with the optimization results, where the training inputs are defined as: $$X_{(\alpha )}=[F_{(\alpha )}(x_1),F_{(\alpha )}(x_2),\dots ,F_{(\alpha )}(x_n)]\alpha =1,2,\dots ,n_{spectrum}$$ (22) and the training outputs are defined as: $$Y_{(\alpha )}=P_{(\alpha )}^g\alpha =1,2,\dots ,n_{spectrum}$$ (23)

Online phase

The global optimization result of each online measured spectrum $F$ is inferred by the constructed RBFN, where the input is set as: $$X=[F(x_1),F(x_2),\dots ,F(x_n)]$$ (24)

The output is calculated as: $$Y={\displaystyle \sum _{\beta =1，2，\dots ,N}{\omega }_{\beta }\psi (\Vert X-c_{\beta }\Vert )}$$ (25) where $\psi$ is a basis function, $\Vert •\Vert$ denotes the Euclidean norm, ${\omega }_{\beta }$ is the weight in the output layer, $N$ is the number of neurons in the hidden layer, and $c_{\beta }$ is the center of RBF in the input vector space. SA is performed with the initial solution $Y$, and the initial state $A$ is defined as $Y$. Initial temperature is set as $T$, and the maximum iteration of SA is set as $n_{SA}$.

For $t=1,2,\dots ,n_{SA}$

The cooling condition is that the best state remains unchanged for $K$ times.

While the cooling condition is not satisfied

Use a perturbation mechanism to generate a new state $A^{\prime }$: $$A^{\prime }=A+randn•\Delta$$ (26) where $randn$ is a normally distributed random number. $\Delta$ is defined as: $$\Delta =[{\delta }_1,{\delta }_2,\dots ,{\delta }_{\tau +m}]$$ (27) and ${\delta }_{\gamma }$ ($\gamma =1,2,\dots ,\tau +m$) is defined with a random integer ${\gamma }_0$ in $[1,\tau +m]$: $${\delta }_{\gamma }=\{\begin{array}{cc}1& \gamma ={\gamma }_0\\ 0& \gamma \ne {\gamma }_0\end{array}$$ (28)

The decrease of fitness is: $$dz=z(A^{\prime })-z(A)$$ (29)

Check whether the new state should be accepted according to Metropolis criteria: $$A=\{\begin{array}{cc}{A}^{\prime }& df<0\text{ or }\exp (-\frac{df}{\kappa T})>rand\\ A& \text{else}\end{array}$$ (30) where $\kappa$ is Boltzmann constant and $rand$ is a random number in $[0,1]$.

End

Cool down with a parameter $\rho$: $$T=\rho T$$ (31)

End

The result of $[a_0,a_1,\dots ,a_{\tau },w_1,w_2,\dots w_m]$ is returned to calculate multi-element concentrations.