## 2 Causality Test Between Environmental Factors and Load Data
### Relationship Identification of Load and Influencing Factors Based on QGCT  
Granger causality test is an approach that can be used to analyze the causality between variables. Granger causality between the influencing factor and the load series is defined as the fact that the influencing factor is considered to be the Granger cause of the load change if the effect of the load forecasting under conditions that include past information about the influencing factor and the load series is better than the effect of forecasting from the past information about the load alone, which means that the influencing factor contributes to the explanation of future load changes. Assume that \(I_{t}=(I_{t}^{Y},I_{t}^{Z})^{\prime}\in\mathbb{R}^{d},d=s+q\) is an explanatory vector and assume that the past information set for \(Z_{t}\) is \(I_{t}^{Z}=(Z_{t-1},\ldots Z_{t-q})^{\prime}\in\mathbb{R}^{q}\). Define the null hypothesis of Granger non-causality from \(Z_{t}\) to load series \(Y_{t}\) as follows [31]:  
\[H_{0}^{Z\to Y}:F_{Y}(y\big{|}I_{t}^{Y},I_{t}^{Z})=F_{Y}(y\big{|}I_{t}^{Y}),y \in\mathbb{R} \tag{4}\]  
where \(F_{Y}(\cdot|I_{t}^{Y},I_{t}^{Z})\) is the conditional distribution function given historical load data \((I_{t}^{Y},I_{t}^{Z})\). Define the null hypothesis of (4) as Granger non-causality in the distribution. Since the estimation of the conditional distribution may be complex caused by the nonlinearity and uncertainty in load, the mean Granger non-causality test is proposed, which is a necessary condition for (4). In this case, \(Z_{t}\) is not a Granger cause of \(Y_{t}\):  
\[E(Y_{t}\big{|}I_{t}^{Y},I_{t}^{Z})=E(Y_{t}\big{|}I_{t}^{Y}) \tag{5}\]  
where \(E(Y_{t}\big{|}I_{t}^{Y},I_{t}^{Z})\) and \(E(Y_{t}\big{|}I_{t}^{Y})\) are the load averages of \(F_{Y}(\cdot\big{|}I_{t}^{Y},I_{t}^{Z})\) with historical information on the influencing factors and \(F_{Y}(\cdot\big{|}I_{t}^{Y})\) without historical information on the influencing factors, respectively. The mean Granger non-causality of (5) can be easily extended to higher order moments. However, causation in the mean ignores the possibility of tail dependence in the distributional conditions; on the other hand, the null hypothesis of the Granger non-causal distribution in (4) does not demonstrate the degree of causation under the conditions in which (4) is rejected. Thus, the use of conditional quantiles to test Granger non-causality identifies patterns of causality and provides sufficient conditions for testing the null hypothesis in (4). Let \(Q_{\tau}^{Y,Z}(\cdot\big{|}I_{t}^{Y},I_{t}^{Z})\) denote the \(\tau\)-quantile of \(F_{Y}(\cdot\big{|}I_{t}^{Y},I_{t}^{Z})\), so (4) can be rewritten as:  
\[H_{0}^{QC,Z\to Y}:Q_{\tau}^{Y,Z}(Y_{t}\big{|}I_{t}^{Y},I_{t}^{Z})=Q_{\tau}^{Y }(Y_{t}\big{|}I_{t}^{Y}),\ \tau\in T \tag{6}\]  
where \(T\) denotes a tight set such that \(T\subset[0,1]\), and the conditional \(\tau\)-quantile of load series \(Y_{t}\) satisfies the following restrictions:  
\[ \begin{cases}\Pr\big{\{}Y_{t}\leq Q_{\tau}^{Y}\big{(}Y_{t}\big{|}I_{t}^{Y} \big{)}\big{|}I_{t}^{Y}\big{\}}:=\tau,\tau\in T,\\ \Pr\Big{\{}Y_{t}\leq Q_{\tau}^{Y,Z}\big{(}Y_{t}\big{|}I_{t}^{Y},I_{t}^{Z}\big{)} \big{|}I_{t}^{Y},I_{t}^{Z}\Big{\}}:=\tau,\tau\in T.\end{cases}  \tag{7}\]  
Given an explanatory vector \(I_{t}\), then \(\Pr\big{\{}Y_{t}\leq Q_{\tau}(Y_{t}\big{|}I_{t})\big{|}I_{t}\big{\}}=E\big{\{} f(Y_{t}\leq Q_{\tau}(Y_{t}\big{|}I_{t}))|I_{t}\big{\}}\). \(f(Y_{t}\leq y)\) is an indicator function for events where \(Y_{t}\) is less than or equal to \(y\). Thus, the Granger non-causal null hypothesis of (6) can be equated:  
\[E\Big{\{}f(Y_{t}\leq Q_{\tau}^{Y,Z}(Y_{t}\big{|}I_{t}^{Y},I_{t}^{Z}))\big{|}I_ {t}^{Y},I_{t}^{Z}\Big{\}}=E\Big{\{}f(Y_{t}\leq Q_{\tau}^{Y}(Y_{t}\big{|}I_{t} ^{Y}))\big{|}I_{t}^{Y}\Big{\}},\ \tau\in T \tag{8}\]  
where the left half of the median is equal to the \(\tau\)-quantile of \(F_{Y}(\cdot\big{|}I_{t}^{Y},I_{t}^{Z})\).