# A Perfect Decomposition Model for Analyzing Transportation Energy Consumption in China
## 2 Methodology
### Decomposition Model Construction According to Factor Direction
#### 2.1.1 Two-Factor Decomposition Model  
We take a two-factor model as a sample example to describe this principle. Figure 1 illustrated the process of the factor changes in different directions, i.e., \(x_{1}\) decreases by \(\Delta x_{1}\), and then \(x_{2}\) increases by \(\Delta x_{2}\). Assume that \(v=x_{1}x_{2}\), i.e., variable \(v\) is determined by factors \(x_{1}\) and \(x_{2}\), within the time period \([0,t]\), \(x_{1}^{t}=x_{1}^{0}+\Delta x_{1}\), \(x_{2}^{t}=x_{2}^{0}+\Delta x_{2}\), the change of variable \(\Delta v\) can be represented as:  
\[\Delta v=v^{t}-v^{0}=x_{1}^{t}x_{2}^{t}-x_{1}^{0}x_{2}^{0}=x_{2}^{0}\Delta x_ {1}+x_{1}^{0}\Delta x_{2}+\Delta x_{1}\Delta x_{2} \tag{1}\]  
where \(x_{2}^{0}\Delta x_{1}\) and \(x_{1}^{0}\Delta x_{2}\) represent the contributions of \(x_{1}\) and \(x_{2}\) to the total change of variable \(v\), respectively; \(\Delta x_{1}\Delta x_{2}\) is the residual term. There are two situations need to be discussed. The factors change at the same direction;  
If \(x_{1}\) increases by \(\Delta x_{1}\), \(x_{2}\) increases by \(\Delta x_{2}\), accordingly. The complete decomposition of two factors is as follows:  
\[x_{1-effect}=x_{2}^{0}\Delta x_{1}+\frac{1}{2}\Delta x_{1}\Delta x_{2} \tag{2}\]  
\[x_{2-effect}=x_{1}^{0}\Delta x_{2}+\frac{1}{2}\Delta x_{1}\Delta x_{2} \tag{3}\]  
The term \(\Delta x_{1}\Delta x_{2}\) is the residual term in the traditional decomposition method, which can be divided equally to the contributions of \(x_{1}\) and \(x_{2}\). Both the changes of \(x_{1}\) and \(x_{2}\), i.e., \(\Delta x_{1}\) and \(\Delta x_{2}\), determines the contributions. If one of \(\Delta x_{1}\) and \(\Delta x_{2}\) is zero, the other is also zero. The factors change in different directions;  
Figure 1 illustrated the process of the factor changes in different directions, i.e., \(x_{1}\) decreases by \(\Delta x_{1}\), and then \(x_{2}\) increases by \(\Delta x_{2}\). Figure 1: Index change when two factors change at different directions. It can be seen from Figure 2 that, when \(x_{2}\) increase by \(\Delta x_{2}\), the total change of \(v\) increases by \(x_{1}^{0}x_{3}^{0}\Delta x_{2}\), (\(x_{1}^{0}x_{3}^{0}\Delta x_{2}=x_{1}^{i}x_{1}^{0}\Delta x_{2}+x_{1}^{0}\Delta x_{ 2}\Delta x_{3}\)); when \(x_{1}\) increases by \(\Delta x_{1}\), the total change of \(v\) increases by \(x_{2}^{0}x_{3}^{0}\Delta x_{1}=x_{1}^{i}x_{2}^{0}\Delta x_{1}+x_{2}^{0}\Delta x _{1}\Delta x_{3}\); when \(x_{3}\) decreases by \(\Delta x_{3}\), the total change of \(v\) decreases by \(x_{2}^{0}x_{1}^{0}\Delta x_{3}\), \(x_{2}^{0}\Delta x_{1}\Delta x_{3}\), \(x_{1}^{0}\Delta x_{2}\Delta x_{3}\), and \(\Delta x_{1}\Delta x_{2}\Delta x_{3}\). At the same time, it also cancels out the total change of \(x_{2}^{0}\Delta x_{1}\Delta x_{3}\), \(x_{1}^{0}\Delta x_{2}\Delta x_{3}\), and \(\Delta x_{1}\Delta x_{2}\Delta x_{3}\), because \(x_{2}\) increases by \(\Delta x_{2}\) and \(x_{1}\) increases by \(\Delta x_{1}\). Each factor increases or decreases to offset the other. The contribution of \(x_{1}\), \(x_{2}\) and \(x_{3}\) to the total change of \(v\) is represented in the following formulas, respectively. \[x_{1-effect}=\Delta x_{1}x_{2}^{0}x_{3}^{0}+\frac{\Delta x_{1}x_{3}^{0}\Delta x _{2}}{2} \tag{10}\]  
\[x_{2-effect}=\Delta x_{2}x_{1}^{0}x_{3}^{0}+\frac{\Delta x_{2}x_{3}^{0}\Delta x _{1}}{2} \tag{11}\]  
\[x_{3-effect}=x_{1}^{0}x_{2}^{0}\Delta x_{3}+\Delta x_{2}\Delta x_{3}x_{1}^{0}+ \Delta x_{1}\Delta x_{3}x_{2}^{0}+\Delta x_{1}\Delta x_{2}\Delta x_{3} \tag{12}\]  
It can be seen that, when the factors change at different directions, an amount of changes are negative, such as \(x_{2}^{0}x_{1}^{0}\Delta x_{3}\), \(x_{2}^{0}\Delta x_{1}\Delta x_{3}\), \(x_{1}^{0}\Delta x_{2}\Delta x_{3}\), the residual \(\Delta x_{1}\Delta x_{2}\Delta x_{3}\) belongs to the change of the negative factors. The change of two factors is negative, and one factor is positive;  
Assume that the change of \(x_{1}\) and \(x_{2}\) are negative, while the change of \(x_{3}\) is positive.