## 4 Electromagnetic Exposure Safety Assessment and Uncertainty Quantification
### Uncertainty Quantification  
In order to quantify the uncertainty of the electromagnetic radiation from mobile phones to the human head, this paper mainly studied the SAR value of the human head when the human body was exposed to the electromagnetic radiation from a mobile phone. Considering the lack of an accurate understanding of the electromagnetic material parameters of the modeled mobile phone, random variables usually exist to generate uncertainty. The distribution type, specific distribution interval, and some parameters are shown in Table 3: \begin{table}
\begin{tabular}{c c c} \hline \hline
**Dielectric Constant** & **Distribution Pattern** & **Distribution Interval** \\ \hline camera & uniform & [1.8,2.0] \\ battery jar & uniform & [4.8,5.2] \\ inner shell & normal & [2.3,0.52] \\ shell & uniform & [2.0,2.4] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Variables of the mobile phone electromagnetic radiation safety assessment for the human head. Figure 7: SAR results at four frequencies. The chosen frequencies were (**a**) 2100 MHz, (**b**) 2300 MHz, (**c**) 3500 MHz, (**d**) 5200 MHz. The Latin hypercube sampling (LHS) is a method to provide a sample of random variables for the MC method, and it is commonly believed that LHS and its modifications can reduce the sample size of the MC method and thus greatly improve its efficiency [31]. The simulation results were obtained by sampling the 1000 runs of the MC method with the LHS. After the construction of the generalized chaotic polynomial was completed, the coefficient matrix was obtained by the random response surface method, and the response value matrix of the system was obtained by multiplying the sample matrix by the coefficient matrix. Then, the statistical characteristics of the system were drawn; the mean and standard deviation are shown in Table 4:  
As can be seen from the comparison results shown in Table 4, the analysis results of the gPC method and the MC method showed good consistency in the overall trend and order of magnitude; so, it was proved that the gPC method is effective for the evaluation of the electromagnetic radiation to the human head caused by mobile phones. The probability density function (PDF) curve obtained by the gPC method was combined with the reference PDF curve drawn by the MC method. Figure 8 shows the results of the gPC calculations and the comparative analysis with the MC method. As shown, the correctness and accuracy of the gPC method was further verified. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Frequency**} & \multicolumn{2}{c}{**Output Mean**} & \multicolumn{2}{c}{**Standard Deviation**} \\  & **MC** & **gPC** & **MC** & **gPC** \\ \hline
2.1 GHz & 0.9576 & 0.9580 & 0.0274 & 0.0277 \\
2.3 GHz & 1.4436 & 1.4431 & 0.0159 & 0.0165 \\
3.5 GHz & 1.7621 & 1.7622 & 0.0511 & 0.0501 \\
5.2 GHz & 1.7737 & 1.7731 & 0.0119 & 0.0117 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The comparison between the output mean and the standard deviation of the SAR. Figure 8: Comparison of the probability density function of the SAR at four frequencies, i.e., (a) 2100 MHz, (b) 2300 MHz, (c) 3500 MHz, (d) 5200 MHz. From the comparison results shown in Figure 8, it can be seen that the PDF of the SAR value analysis results of the gPC and the MC methods were consistent at different frequencies. It was demonstrated that the gPC method was highly accurate in quantifying the uncertainty of the electromagnetic radiation from mobile phones to the human head. The confidence interval is also a statistical quantity. It refers to the use of a sample statistic to estimate an overall parameter given a range such that the probability of the overall parameter falling within that range is at a pre-determined confidence level. In this paper, the mean of the sample was used to estimate the overall mean, but due to the random nature of the sample, we could not guarantee that the estimate would be exactly the same as the true overall mean.