# Empirical Evaluation on Utilizing CNN-Features for Seismic Patch Classification
## 2 Related Work
### Data Complexity  
An attempt to identify the relationship between the data and the classifier that classifies it was specifically initiated in [34]. Since then, it has been studied a lot. A few examples, but not all, can be found in [13; 35; 36]. Referring to these findings, we selected the data complexity measures to be used in our experiments as the following four indicators: the Fisher's discriminant ratio (\(F1\)); the directional-vector Fisher's discriminant ratio (\(F1v\), or simply, \(Fv\)); the distance of erroneous instances to a linear classifier, or its training error rate (\(L2\)); and the volume of the local neighborhood (\(D2\)).  
Each measurement of complexity mentioned above is made as follows. First, \(F1\) is used to measure the separation capacity of a single feature between two classes. One way to obtain the value for multiple features is:  
\[F1=\sum_{i=1}^{2}p_{i}(\mathbf{m}-\mathbf{m}_{i})\mathbf{C}^{-1}(\mathbf{m}-\mathbf{m}_{i}), \tag{1}\]  
where \(\mathbf{m}_{i}\) (and \(\mathbf{m}\)) is the mean of each class (and \(c_{i}\) the entire class); \(\mathbf{C}\) is the pooled covariance matrix derived from averaging the covariance of each class; and \(p_{i}\) is the proportion of examples in the class, \(c_{i}\). As a result, the higher the value, the less redundancy and the easier it is to distinguish between the two classes.  
Second, the \(Fv\) value, developed as a complement to \(F1\), is calculated as in [29]:  
\[Fv=\frac{1}{1+\mathbf{d}^{T}\mathbf{B}\mathbf{d}/\mathbf{d}^{T}\mathbf{W}\mathbf{d}}, \tag{2}\]  
where \(\mathbf{B}\) is the between-class scatter matrix; \(\mathbf{W}\) is the within-class scatter matrix; and \(\mathbf{d}=\mathbf{W^{-1}}(\mathbf{m_{1}}-\mathbf{m_{2}})\). That is, \(\mathbf{d}\) is a directional vector in which the data are projected to maximize class separation, and the value of \(Fv\) approaches zero by maximizing \(\mathbf{W^{-1}}\mathbf{B}\), which becomes a simpler classification. Therefore, the lower the measurement, the simpler the classification problem is.  
Third, \(D2\) is related to the density measurement, defined as the average number of samples per unit volume in the space where all samples are distributed [36]. The value of \(D2\) in \(n\) training samples is determined by measuring the average volume occupied by the \(k\) nearest neighbors in each sample, \(x_{i}\), referred to as \(N_{k}(x_{i})\). The value is counted as:  
\[D2=\frac{1}{n}\sum_{i=1}^{n}\prod_{h=1}^{d}(max(f_{h},N_{k}(x_{i}))-min(f_{h}, N_{k}(x_{i})\ )), \tag{3}\]  
where \(max(f_{h},N_{k}(x_{i}))\) and \(min(f_{h},N_{k}(x_{i})\ )\) denote the maximum and minimum values of the feature, \(f_{h}\), among the k-NN of \(x_{i}\), respectively.  
Finally, the \(L2\) value is decided by referring to the error rate of the linear SVM classifiers. Therefore, the higher the value, the more errors, and the more difficult it is to classify linearly, which increases the complexity. Here, each measurement is briefly described to the minimum required. A detailed description of each of these measurements, and other measures that were not selected here, but that were closely related to the selected ones, can be found in the relevant literature.