# A Comprehensive Survey for Deep-Learning-Based
## 5 Methodologies
### Data-Driven Modeling for the Power System  
In the era of big data [42; 101], the data-driven methods provide a new kind of solution for reliable WAMPAC. For example, [PERSON] etc. proposed a robust principal component analysis (PCA)-based abnormal detection method [6]. The singular value decomposition \begin{table}
\begin{tabular}{l l l} \hline \hline
**Visible Light Data Resource** & **Advantages** & **Shorcomings** \\ \hline Ground patel & High detection rate & Slow, tedious, dangerous for human, labor-intensive, impossible in extreme weather conditions and harsh terrains \\ \hline Fixed camera & Low maintenance & Limited observation range, complex deployment, high cost \\ \hline Climbing rebet & High inspection accuracy & Slow, weight could damage the lines, hard to pass across various obstacles \\ \hline Helicopter & Fast inspection speed, wide observation range & High cost, safety issues \\ \hline Unmanned Aerial & Automatic or human-controlled, real-time, safe, fly close to the detection target & Difficult for power line tracking and navigation \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Platforms For collecting visible light data. (SVD)-based indicator was advanced by the work of [40]. Inspired by random matrix theories, the linear eigenvalue statics of considered PMU data were proposed by [7]; however, this brought a massive computational expense due to the high dimensionality of the employed data. Recently, based on the high-dimensional covariance matrices test, the work of [50] proposed a low-complexity algorithm to address this challenging issue, which works well in both a small- and large-scale power system. Very recently, using random matrix theory [49; 52; 54; 102], [PERSON] et al. proposed an improvement of power system state estimation by presenting matrix cleaning (MC) technology [44]. Instead of numerically introducing the details of these methods, we provide insightful results below, with which one can understand the technical connections with the classical methods. Taking the trace operator of the Equation (2), we can get  
\[\mathbf{Tr(A)}\mathbf{=}\mathbf{Tr}\Big{(}\mathbf{B}^{T}\mathbf{C}\Big{)}. \tag{3}\]  
Denoting the normalized trace operator as \(\mathbf{tr(\cdot)}=\frac{1}{N}\mathbf{Tr(\cdot)}\), then we can have  
\[\frac{1}{N}\mathbf{Tr}\Big{(}\mathbf{AA}^{T}\Big{)}\mathbf{=}\frac{1}{N} \mathbf{Tr}\Big{(}\mathbf{B}^{T}\mathbf{CC}^{T}\mathbf{B}\Big{)}. \tag{4}\]  
It is noted that the trace operator obeys the following property:  
\[\mathbf{tr(XYZ)}=\mathbf{tr(ZXY)}, \tag{5}\]  
\[\mathbf{tr}\Big{(}\mathbf{X}^{T}\mathbf{Y}\Big{)}\leq\Big{[}\mathbf{tr}\Big{(} \mathbf{X}^{T}\mathbf{X}\Big{)}\Big{]}^{1/2}\Big{[}\mathbf{tr}\Big{(}\mathbf{ Y}^{T}\mathbf{Y}\Big{)}\Big{]}^{1/2}, \tag{6}\]  
Combining (4)-(6), we can obtain  
\[\mathbf{tr}\Big{(}\mathbf{AA}^{T}\Big{)}=\mathbf{tr}\Big{(}\mathbf{B}^{T} \mathbf{CC}^{T}\mathbf{B}\Big{)}=\mathbf{tr}\Big{(}\mathbf{BB}^{T}\mathbf{CC} ^{T}\Big{)}, \tag{7}\]  
and  
\[\mathbf{tr}\Big{(}\mathbf{AA}^{T}\Big{)}\leq\Big{[}\mathbf{tr}\Big{(}\mathbf{ B}^{T}\mathbf{BBB}^{T}\Big{)}\Big{]}^{1/2}\Big{[}\mathbf{tr}\Big{(}\mathbf{ C}^{T}\mathbf{CCC}^{T}\Big{)}\Big{]}^{1/2}. \tag{8}\]  
Let \(\sigma_{\mathbf{max}}\) be the maximum singular value of \(\mathbf{B}\), and \(\sigma_{\mathbf{max}}\) satisfies  
\[\sigma_{\mathbf{max}}^{2}\geq\mathbf{tr}\Big{(}\mathbf{B}^{T}\mathbf{BBB}^{ T}\Big{)}, \tag{9}\]  
From (7)-(9), we can get  
\[\mathbf{tr}\Big{(}\mathbf{AA}^{T}\Big{)}\leq\sigma_{\mathbf{max}}\Big{[} \mathbf{tr}\Big{(}\mathbf{C}^{T}\mathbf{CCC}^{T}\Big{)}\Big{]}^{1/2}. \tag{10}\]  
Base on the analysis from (1) and (10), the normalized trace operator of \(\mathbf{AA}^{T}\) can be equivalently regarded as a novel indicator of abnormality detection in smart grids. The above simple analysis can bring critical insight to the basics of data-driven methods. In conclusion, data-driven abnormal detection can provide reasonable solutions with high-dimensional data [44; 50; 103; 104].