## 3 Experimental Results and Discussion
### Results and Discussion of Deep Learning Models  
The MLE, as we know, is the best estimator for 2D FBM--it is an unbiased estimator and has the lowest mean-squared error. Even so, when its estimates are used to differentiate between the classes of the Hurst exponent or fractal dimension, the classification rate of the SVM classifier on these estimates is only 83.9% for FBSs and 81.4% for FBIs under the resolution of \(1/11\), and only 56.0% for FBSs and 53.2% for FBIs under the resolution of \(1/21\). For images of size \(32\times 32\times 1\), the resolution is not very high, especially under the resolution of \(1/21\). Therefore, the first purpose of the paper is to investigate whether or not we can design a simple sequential network to achieve a higher classification rate, as well as whether some pretrained deep learning models can obtain higher classification rates. If so, the second purpose is to use these deep learning models to replace the role of the MLE for 2D FBM in order to efficiently and effectively recognize the classes of images--or equivalently estimate the Hurst exponent or its corresponding fractal dimension of images. First, we designed a simple sequential network model and chose three pretrained models--AlexNet, GoogleNet, and Xception--as our compared subjects. Except for the AlexNet model, each model contains two types of modalities: one uses the size of the original images--32 \(\times\) 32 \(\times\) 1--as the size of the input layer, and hence no augmented operation is necessary; the other uses the size of the original model--for example, 227 \(\times\) 227 \(\times\) 3 for the AlexNet model--as the size of the input layer, and hence an augmented operation is needed. We call the first modality Type 1 and the second Type 2 and simply express them as AlexNet 1 and AlexNet 2, GoogleNet 1 and GoogleNet 2, as well as Xception1 and Xception 2. Since the AlexNet model contains three pooling layers, making the size smaller and smaller, AlexNet 1 was not suitable for the size of our images. Therefore, the following four tables will not list their corresponding results. Totally, six models were considered and performed. All six models were run on the same FBIs or 2D FBM images and under five-fold cross-validation. The size of our images is 32 \(\times\) 32 \(\times\) 1. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Classes** & **Precision** & **Recall** & **F1-Score** & **Support** \\ \hline
1 & 1.00 & 1.00 & 1.00 & 200 \\
2 & 0.97 & 0.96 & 0.97 & 200 \\
3 & 0.91 & 0.93 & 0.92 & 200 \\
4 & 0.84 & 0.89 & 0.87 & 200 \\
5 & 0.85 & 0.80 & 0.82 & 200 \\
6 & 0.79 & 0.80 & 0.79 & 200 \\
7 & 0.79 & 0.78 & 0.78 & 200 \\
8 & 0.73 & 0.76 & 0.74 & 200 \\
9 & 0.67 & 0.70 & 0.68 & 200 \\
10 & 0.61 & 0.66 & 0.63 & 200 \\
11 & 0.79 & 0.67 & 0.73 & 200 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The classification report of SVMs on the estimates of FBIs of \(11\) Hurst exponents. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**TC a\({}^{\text{b}}\)** & **PC b** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** \\ \hline
1 & 200 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
2 & 1 & 192 & 7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
3 & 0 & 5 & 185 & 10 & 0 & 0 & 0 & 0 & 0 & 0 \\
4 & 0 & 0 & 12 & 178 & 10 & 0 & 0 & 0 & 0 & 0 \\
5 & 0 & 0 & 0 & 23 & 159 & 18 & 0 & 0 & 0 & 0 \\
6 & 0 & 0 & 0 & 19 & 159 & 22 & 0 & 0 & 0 & 0 \\
7 & 0 & 0 & 0 & 0 & 0 & 24 & 155 & 21 & 0 & 0 & 0 \\
8 & 0 & 0 & 0 & 0 & 0 & 19 & 151 & 30 & 0 & 0 \\
9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 35 & 140 & 25 & 0 \\
10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 33 & 132 & 35 \\
11 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 59 & 134 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The confusion matrix of SVMs on the estimates of FBIs of 11 Hurst exponents.