## 3 Results and Discussions
### Most Sensitive Spectral Band
With the optimal network of seven layers which is shown in Figure 5, we also performed deep learning method on other bands of the spectral range with the same data set to test whether spectral band of 12.5 to 16.5 MHz really leads to the most accurate evaluation of average adipocyte size among the different spectral bands. The spectral range of 0.5 to 24.5 MHz was divided into twenty-one spectral bands. Each spectral band had a length of 4 MHz as well as a 3 MHz (75%) overlap with the adjacent spectral band. For each of the twenty-one spectral bands, the normalized magnitude of the 4-MHz spectral band worked as the input of the deep network, while the corresponding average adipocyte size worked as the output. Using data augmentation illustrated in Section 3.2, the deep network was trained for each band. Then, the relationship found out from the training was tested using the test set. For each of the 4-MHz spectral bands, the network was retrained for 10 times, and the average error of each time was computed. The average MRE from the 10 retrainings over different spectral bands are listed in Table 2.  
Figure 4: Training and testing using the spectral band of approximately 12.5 to 16.5 MHz with different layer numbers. The minimum, maximum, and average of MREs of test set for 10 retrainings are drawn for each layer number. Figure 5: The structure of deep network. The numbers of the elements of the input and the output are marked on the left of each layer in the form of “\({I_{m}}/{O_{m}}\)” for the \(m\)-th layer. All the layers are connected in the same as shown in Figure 1. The details of layer 1 and layer 2 are presented in the right as an example. By studying each 4-MHz spectral band in the entire spectral band using the deep network, the spectral band of 12.5 to 16.5 MHz turned out to be mostly sensitive to the adipocyte size. As is shown in Figure 6, in the range of 0.5 to 16.5 MHz, spectral bands covering higher frequencies lead to smaller errors; while in the range of 16.5 to 24.5 MHz, the error becomes larger when the frequency is higher. In addition to changes in MREs, the range of errors also changes with the spectral bands as well. In the spectral band of 0.5 to 4.5 MHz, the error varies in a large range. The range of errors decreases when the spectral band moves from 0.5 to 4.5 MHz to 12.5 to 16.5 MHz. However, when the spectral bands changes from 12.5 to 16.5 MHz to 20.5 to 24.5 MHz, the range of errors also increases. As a result, the spectral range of 12.5 to 16.5 MHz leads to the minimum error, and also a much smaller range of errors. The variation of error in different spectral bands results from the adipocyte size distribution. As illustrated in Figure 2d, we used Gamma distribution to fit the adipocyte size distribution. Number of adipocytes with sizes close to average size is larger than that of adipocytes whose size is away from the average. Therefore, the spectral band which is closer to the most sensitive spectral band corresponds to adipocytes accounting for a larger proportion and makes the performance of deep learning network more accurate and stable. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Spectral Band/MHz** & **Average of MRE** & **Spectral Band/MHz** & **Average of MRE** \\ \hline
0.5–4.5 & 19.04\% & 11.5–15.5 & 13.27\% \\
1.5–5.5 & 18.59\% & **12.5–16.5** & **12.78\%** \\
2.5–6.5 & 18.53\% & 13.5–17.5 & 13.93\% \\
3.5–7.5 & 16.48\% & 14.5–18.5 & 14.50\% \\
4.5–8.5 & 16.73\% & 15.5–19.5 & 14.16\% \\
5.5–9.5 & 14.82\% & 16.5–20.5 & 15.15\% \\
6.5–10.5 & 14.10\% & 17.5–21.5 & 16.41\% \\
7.5–11.5 & 14.41\% & 18.5–22.5 & 16.78\% \\
8.5–12.5 & 13.85\% & 19.5–23.5 & 16.02\% \\
9.5–13.5 & 13.79\% & 20.5–24.5 & 16.10\% \\
10.5–14.5 & 13.92\% & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Errors from the test set over different spectral bands. Figure 6: Plot showing mean relative error (MRE) of test set and the corresponding spectral bands. For each spectral band, the minimum, maximum and average of MREs of test set for 10 retrainings are drawn at the center frequency.