## 2 Methodology
### Correlation Analysis  
As a basic method in statistics, correlation analysis is used to dig out the relationship between a pair of variables. For predicting future behavior, it is essential that the variables have the similar trend or distribution. An expression called the Pearson correlation coefficient (PCC) is commonly recognized as the standard of correlation analysis. The value of PCC ranges from \(-1\) to \(1\), with \(-1\), \(0\), \(1\) indicating that the two variables are perfectly negatively correlated, uncorrelated and perfect positive correlated, respectively. An autocorrelation coefficient (AC) is a special case of PCC where the variables are the sub-series of the same series with various time series indexes. By defining \(\mathsf{S}(t_{0})\) and \(\mathsf{S}(t_{1})\) as the sub-series of variable \(\mathsf{S}\) with time series indexes \(t_{0}\) and \(t_{i}\) (\(i\
eq 0\)), the AC is defined as  
\[\rho_{\mathsf{S}(t_{i})}=\frac{Cov\left(\mathsf{S}(t_{0}),\mathsf{S}(t_{i}) \right)}{\sqrt{D\left(\mathsf{S}(t_{0})\right)}\sqrt{D\left(\mathsf{S}(t_{i}) \right)}}=\frac{Cov\left(\mathsf{S}(t_{0}),\mathsf{S}(t_{i})\right)}{D\left( \mathsf{S}\right)}, \tag{1}\]  
where \(\rho_{\mathsf{S}(t_{i})}\) is the AC between variables \(\mathsf{S}(t_{0})\) and \(\mathsf{S}(t_{1})\), \(Cov\left(\cdot\right)\) is the covariance, and \(D\left(\cdot\right)\) is the variance. \(\rho_{\mathsf{S}(t_{i})}\) is determined by \(t_{i}\) only. For both PCC and AC, it is generally accepted that a little correlation exists when their absolute values are below \(0.3\) and the absolute values exceed \(0.5\) means that the series are related [41]. Figure 1: Procedure for signal decomposition using wavelet packet transform (WPT). In this paper, the two above standards are adopted to pick out the relevant sub-series from load data and extract features from the wavelet components, which have stronger autocorrelation than the origin sub-series. The proposed method is shown in Figure 2.  
According to Figure 2, the load data \(S\) is divided into load features \(F\) and noise \(N\). \(\rho_{1}\) and \(\rho_{2}\) are the standards for load data filtering in time domain and wavelet domain. \(A^{*}(\mathbf{t})\), \(\mathbf{D}^{*}(\mathbf{t})\) and \(A^{\sim}(\mathbf{t})\), \(\mathbf{D}^{\sim}(\mathbf{t})\) are the sub-series of wavelet components at time series \(\mathbf{t}\). To be proportional to correlation, ACs are in the form of absolute value. Indeed, the PCCs between different approximation and detail components are near zero since the components are decomposed by orthogonal wavelet bases. Then, Equation (1) is also expressed as  
\[ \begin{split}\rho_{S(t_{i})}&=\frac{Cov\left(S(t_{ 0}),S(t_{i})\right)}{\sqrt{D\left(S(t_{0})\right)}\sqrt{D\left(S(t_{i})\right)} }=\frac{Cov\left(\mathbf{F}(t_{0})+\mathbf{N}(t_{0}),\mathbf{F}(t_{i})+\mathbf{N}(t_{i}) \right)}{\sqrt{D\left(\mathbf{F}(t_{0})+\mathbf{N}(t_{0})\right)}\sqrt{D\left(\mathbf{F}( t_{i})+\mathbf{N}(t_{i})\right)}}\\ &=\frac{Cov\left(\mathbf{F}(t_{0}),\mathbf{F}(t_{i})\right)+Cov\left(F(t_ {0}),\mathbf{N}(t_{i})\right)+Cov\left(\mathbf{N}(t_{0}),\mathbf{F}(t_{i})\right)+Cov\left( \mathbf{N}(t_{0}),\mathbf{N}(t_{i})\right)}{\sqrt{D\left(\mathbf{F}(t_{0})\right)+D\left( \mathbf{N}(t_{0})+2 Cov\left(\mathbf{F}(t_{0}),\mathbf{N}(t_{0})\right)\right)\cdot\sqrt{D \left(\mathbf{F}(t_{i})\right)+D\left(\mathbf{N}(t_{i})\right)+2 Cov\left(\mathbf{F}(t_{i}),\mathbf{N}(t_{i})\right)}}}\\ &\approx\frac{Cov\left(\mathbf{F}(t_{0}),\mathbf{F}(t_{i})\right)+Cov \left(\mathbf{N}(t_{0}),\mathbf{N}(t_{i})\right)}{D\left(\mathbf{F}(t_{0})\right)+D\left( \mathbf{N}(t_{0})\right)}.\end{split}