## 2 Motivation and Preliminary  
### Motivation  
Wind turbine blade-cracking detection is typically a binary classification problem. In order to explain the effects of imbalanced data, this section takes two common binary classification methods for analysis, namely decision tree (C4.5) and support vector machine (SVM). For the former, a splitting node is selected according to the information gain ratio:  
\[ \begin{split}\text{Gain\_ratio}&=\frac{\sum\limits_{i =1}^{N}\frac{D_{i}}{D}\Big{(}-\frac{D_{i}^{+}}{D_{i}^{+}}\text{log}_{2}\frac{D _{i}^{+}}{D_{i}^{+}}-\frac{D_{i}^{-}}{D_{i}^{-}}\text{log}_{2}\frac{D_{i}^{- }}{D_{i}^{-}}\Big{)}}{\left(-\frac{D_{i}^{+}}{D}\text{log}_{2}\frac{D_{i}^{+}} {D}-\frac{D_{i}^{-}}{D}\text{log}_{2}\frac{D_{i}^{+}}{D}\right)}\\ &\text{s.t.}\ D=D^{+}+D^{-},D_{i}=D_{i}^{+}+D_{i}^{-}\\ &\ D^{+}\gg D^{-},D_{i}^{+}\gg D_{i}^{-}\end{split}  \tag{1}\]  
where \(D\) denotes the total sample amount; \(D^{+}\) denotes the normal sample amount; \(D^{-}\) denotes the blade cracking sample amount. \(N\) is the total number of attribute values for a specific attribute. \(D_{i}\) denotes the sample amount for \(i\)-th attribute value. \(D_{i}^{+}\) and \(D_{i}^{-}\) are the normal sample amount and cracking sample amount for \(i\)-th attribute value. Since  
\[ \begin{split}\lim\limits_{x\to 0^{+}}x\text{log}_{2}(x)& =\lim\limits_{x\to 0^{+}}x\frac{\text{ln}(x)}{\text{ln}2}=\frac{1}{ \text{ln}2}\lim\limits_{x\to 0^{+}}x\text{ln}(x)\\ &=\frac{1}{\text{ln}2}\lim\limits_{x\to 0^{+}}\frac{\text{ln}(x)}{ \text{ln}^{\text{1/x}}}=\frac{1}{\text{ln}2}\lim\limits_{x\to 0^{+}}\frac{ \text{1/x}}{-\text{1/x}^{2}}=-\frac{1}{\text{ln}2}\lim\limits_{x\to 0^{+}}x=0 \end{split}  \tag{2}\]  
Supposing there are very few cracking samples, we can infer that  
\[\text{Gain\_ratio}\ \approx\frac{\sum\limits_{i=1}^{N}\frac{D_{i}}{D}\bigg{(} \frac{D_{i}^{+}}{D_{i}}\text{log}_{2}\bigg{(}\frac{D_{i}^{+}}{D_{i}^{-}} \bigg{)}\bigg{)}}{\left(\frac{D^{+}}{D}\text{log}_{2}\bigg{(}\frac{D_{i}^{+}} {D}\bigg{)}\right)}=\frac{\sum\limits_{i=1}^{N}\frac{D_{i}^{+}}{D^{+}}\text{ log}_{2}\bigg{(}\frac{D_{i}^{+}}{D_{i}}\bigg{)}}{\text{log}_{2}\bigg{(}\frac{D _{i}^{+}}{D}\bigg{)}} \tag{3}\]  
From Equation (3), it is clear that gain ratio is irrelevant with cracking samples. That is, building the decision tree only depends on normal samples, causing a high recognition uncertainty for cracking samples. For SVM, we take the linear separable case for explanation. If the data are balanced, the decision surface is obtained by minimizing Equation (4):  
\[\underset{\mathbf{w}}{\text{argmin}}\ \text{Loss}\ =\parallel\mathbf{W}\parallel+C\sum \limits_{i=1}^{D}\xi_{i},\ s.t.\ y_{i}(\mathbf{W}\mathbf{x}_{i}+b)\geq 1-\xi_{i},\xi_{i}\geq 0 \tag{4}\]  
where \(\mathbf{W}\) decides the position and direction of decision surface; \(C\) is a penalty coefficient; \(\xi_{i}\) is a relaxation variable; \(\{\mathbf{x}_{i}\), \(y_{i}\}\) denotes \(i\)-th sample. If some cracking samples are removed and surface shifts distance d towards cracking samples, the variation in _Loss_ is computed by Equation (5):  
\[ \begin{split}\Delta Loss&=C\sum\limits_{i=1}^{D^{+}} \left(\xi_{i}-dsin\theta\right)+C\sum\limits_{j=1}^{D^{-}}\left(\xi_{j}+dsin \theta\right)-\left(C\sum\limits_{i=1}^{D^{+}}\xi_{i}+C\sum\limits_{j=1}^{D^{-} }\xi_{j}\right)\\ &=Cdsin\theta(D^{-}-D^{+})<0\end{split}  \tag{5}\]  
where \(\theta\) represents the angle between the x-axis and decision surface. Obviously, \(\Delta Loss\) is less than zero, implying a better support for normal samples, but it may bring many misclassifications of these unseen cracking samples near the former decision surface. From the above analysis, the root cause for low recognition of cracking samples is the imbalanced sample amount. Current solutions mainly include two aspects, namely sampling strategies and generative strategies. However, there are still critical problems that deserve deep research.