# Research on Generalized Hybrid Probability Convolutional Neural Network
## 5 Experiment and Results
### Parameter Analysis  
In the experiment of the generalizability to unknown categories, the parameter \(A\) of a potential function [30] in the GHP-CNN represents the distance weighting factor between the vectors. By adjusting this parameter, the potential value between the two vectors can be changed. When the potential value between two vectors is assumed to be a fixed value, the larger the distance between the two vectors, the smaller \(A\) is. When the distance between two vectors is assumed to be a fixed value, the larger \(A\) is, the smaller the potential value between the sample vectors. Figure 6 shows that when parameter \(A\) changed in a small range, the change in classification accuracy was relatively stable, which showed that the algorithm had a good robustness to parameter \(A\).  
The size of the initial kernel width \(\sigma\) affects the coverage of pattern nodes of the GHP-CNN, thereby affecting the number of mode layer nodes. If the value of \(\sigma\) is too small, then the coverage region of the generated pattern node does not contain other sample points, and the PNN classifier is close to the nearest neighbor classifier, which is equivalent to using a training sample to represent a pattern node, and the structure of the network pattern layer cannot be simplified in this case. If the value of \(\sigma\) is too large, the selected pattern node cannot reflect the real sample distribution and may increase the number of heterogeneous samples in the kernel, which seriously affects the classification accuracy. In the algorithm of this paper, the input sample was a vector after the hyperspherical mapping, so that \(\sigma\) could be kept in a small range for proper selection. As shown in Figure 7, when \(\sigma\) was between 0.05 and 0.25, the GHP-CNN algorithm could achieve a relatively good classification accuracy.  
Note that different \(h\) values lead to differences in the separability of the normalized samples. We can select \(h\) according to the following separability criterion.  
\[h=\operatorname*{arg\,max}_{h}[\frac{tr(\mathbf{SM}_{b})}{tr(\mathbf{SM}_{t})}] \tag{10}\]  
where \(\mathbf{SM}_{b}\) and \(\mathbf{SM}_{t}\) denote the between-class scatter matrix and total scatter matrix of the normalized training samples, respectively. However, as a new dimension, the value of \(h\) should be matched with the other dimensions. Therefore, the initial value of h can be selected as follows.  
\[h=a(\prod\limits_{w=1}^{n}\Delta_{w})^{\frac{1}{h}}+(1-a)(\det(\mathbf{SM}_{t} ))^{\frac{1}{h}} \tag{11}\]  
where \(a\in[0,1]\). \(\prod\limits_{w=1}^{n}\Delta_{w}\) denotes the volume of an n-dimension hypercube for the target sample space, and \(\det(\mathbf{SM}_{t})\) measures the volume of an n-dimension superellipsoid of the sample scatter for the target set. We compute the two volumes differently because the actual distribution shape of the features of the target set is generally irregular.  
To ensure fairness in the comparison, all algorithms were tested using the same CPU. The unit of time consumed was in seconds. As shown in Table 6, the computational efficiency of the CNN-ELM and CNN-SVM algorithms was significantly higher than that of the other algorithms. The VGG-16-E algorithm took the most time because the node parameters of the MLP layer were much larger than in the other classifiers. The time required for the GHP-CNN algorithm proposed in this paper was much lower than the VGG-16-E test time.