## 3 Experimental Results and Discussion
### Results and Discussion of Deep Learning Models
In the case of Type 1, since our image size is different from those of the three pre-trained models--227 \(\times\) 227 \(\times\) 3 for the AlexNet model (25 layers), 224 \(\times\) 224 \(\times\) 3 for the GoogleNet model (144 layers), and 299 \(\times\) 299 \(\times\) 3 for the Xception model (170 layers)--and the number of the output classes (11 or 21) is different from 1000 for three models, we adjusted the input layer, fully connected layer, and an output layer of the three pretrained models in order to run these models correctly. For convenience, when we refer to these adjusted models, we call them the adjusted AlexNet model, the adjusted GoogleNet model, and the adjusted Xception model, respectively, or simply AlexNet 1, GoogleNet 1, and Xception 1, respectively. In the case of Type 2, we used an augmenter to make the image size (32 \(\times\) 32 \(\times\) 1) match the sizes (227 \(\times\) 227 \(\times\) 3, 224 \(\times\) 224 \(\times\) 3, or 299 \(\times\) 299 \(\times\) 3) of three pretrained models, and modified the corresponding fully-connected layer and output layer to meet the number of our classes, 11 or 21. We simply call them AlexNet 2, GoogleNet 2, and Xception 2, respectively. The augmenter was introduced only for resizing the original images to the corresponding sizes of three pretrained models. For a fair comparison, all six models were performed in the same computing environment. (1) Hardware: a computer of Intel(r) Xeon(R) W-2235 CPU @ 3.80 GHz with 48.0 GB RAM (47.7 GB available), together with a GPU processor (NVIDIA RTX A4000); (2) operating system: Windows 11 Professional Workstation, version 21 H2; (3) programming software: MATLAB R2022a; (4) training options: the solver was stochastic gradient descent with momentum (SGDM); the mini-batch size was 128; the initial learning rate was 0.001; the learning rate schedule was piecewise; the learning rate drop factor was 0.1; the learning rate drop period was 20; the shuffle was for every epoch; the validation frequency was 30; the number of epochs was 30; and the other options were set to default. Since the mini-batch size was 128 and five-fold was performed, the number of iterations per epoch was 68 (11,000 \(\times\) 0.8/128 = 68.75) for 11 Hurst exponents, 2040 iterations in total; the number of iterations per epoch was 131 (21,000 \(\times\) 0.8/128 = 131.25) for 21 Hurst exponents, 3930 iterations in total. Our proposed model is a 29-layer sequential network composed of an image input layer, five groups of a convolutional layer, a batch normalization layer, a ReLU layer, and a maximum pooling layer (in total, 20 layers), one group of a convolutional layer, a batch normalization layer, and a ReLU layer (in total, three layers), one group of a fully connected layer and a ReLU layer (in total, two layers), as well as the final group of a fully connected layer, a softmax layer, and a classification layer (in total, three layers). According to the structures and characteristics of our FBIs, we adopted a simple design concept for our proposed model; that is, the filters were regularly configured, and their corresponding sizes were increased. Therefore, our proposed model was arranged as follows. The image input layer is of size 32 \(\times\) 32 \(\times\) 1; the first convolutional layer contains 128 filters of size 3 \(\times\) 3; the second convolutional layer contains 128 filters of size 5 \(\times\) 5; the third convolutional layer contains 128 filters of size 7 \(\times\) 7; the fourth convolutional layer contains 128 filters of size 9 \(\times\) 9; the fifth convolutional layer contains 128 filters of size 11 \(\times\) 11; the sixth convolutional layer contains 128 filters of size 13 \(\times\) 13. All maximum pooling layers are of size 2 \(\times\) 2 with stride two. The output size of the first fully connected layer is 10 times 11 or 21 (110 or 210); the output size of the second fully connected layer is 11 or 21, depending on the number of classes. For clarity, Table 7 shows the architecture of our proposed model.