## 2 Methodology
### Prediction Model Structure Construction
#### 2.2.1 The Classical Model
If car sales movement direction \(Y=U\) is selected as the pivot, then the model for \(Y=U\) is:  
\[\ln\frac{P(Y=F)}{P(Y=U)}=\mathbf{b}_{F}\cdot\mathbf{x}\text{ and }\ln\frac{P(Y=D)}{P(Y=U) }=\mathbf{b}_{D}\cdot\mathbf{x} \tag{3}\]  
where \(Y\) refers to the random variable of \(y_{i}\) in Equation (2); \(\mathbf{b}_{F}\) and \(\mathbf{b}_{D}\) refer to the set of regression coefficients in Equation (2) associated with car sales movement directions \(F\) and \(D\), respectively (\(\mathbf{b}\) is typically estimated by the maximum likelihood method [33]); \(\mathbf{x}\) refers to the vector of \(x_{i1}\) to \(x_{i3}\) in Equation (2). Then, the probability that \(\mathbf{x}\) belongs to car sales movement directions \(F\) and \(D\) can be expressed as Equation (4):  
\[P(Y=F)=P(Y=U)e^{\mathbf{b}_{F}\cdot\mathbf{x}}\text{ and }P(Y=D)=P(Y=U)e^{\mathbf{b}_{D} \cdot\mathbf{x}} \tag{4}\]  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**No. **} & \multirow{2}{*}{**Month**} & \multicolumn{4}{c}{**BAM**} & \multicolumn{4}{c}{**Lexus**} & \multicolumn{4}{c}{**Mazda**} & \multicolumn{2}{c}{**Benz**} & \multicolumn{2}{c}{**Toyota**} & \multicolumn{2}{c}{**VW**} \\ \cline{2-13}  & & **Sales** & **Label** & **Sales** & **Label** & **Sales** & **Label** & **Sales** & **Label** & **Sales** & **Label** & **Sales** & **Label** \\ \hline
1 & 2014/1 & 1437 & D & 1207 & D & 691 & D & 1815 & D & 2471 & D & 897 & D \\
2 & 2014/2 & 624 & U & 806 & U & 231 & U & 1033 & U & 957 & U & 551 & U \\
3 & 2014/3 & 1294 & F & 971 & F & 477 & F & 1588 & F & 1653 & U & 1053 & F \\
4 & 2014/4 & 1393 & F & 1039 & U & 452 & F & 1692 & D & 1943 & F & 1154 & F \\
5 & 2014/5 & 1284 & U & 1310 & D & 466 & D & 1518 & F & 1783 & U & 1113 & U \\
6 & 2014/6 & 1424 & F & 1131 & U & 319 & U & 1620 & F & 2220 & U & 1414 & D \\
7 & 2014/7 & 1469 & D & 1481 & D & 618 & D & 1616 & D & 2631 & D & 959 & F \\
8 & 2014/8 & 1190 & U & 593 & U & 325 & U & 1389 & U & 983 & U & 914 & U \\
9 & 2014/9 & 1454 & U & 1193 & U & 444 & D & 1784 & F & 1615 & F & 1438 & F \\
10 & 2014/10 & 1630 & F & 1334 & D & 298 & U & 1695 & F & 1645 & U & 1313 & D \\
11 & 2014/11 & 1565 & U & 1094 & F & 434 & U & 1754 & F & 2031 & U & 795 & U \\
12 & 20 \begin{table}
\begin{tabular}{c c c c c c} \hline  & & & \multicolumn{3}{c}{**Same-**} & \\
**No** & **Month** & **Historical** & **Month-Last** & **Month** & **Mouth** & **Label** \\
**(i)** & **Month** & **Sales** & **Year Label** & **Number** & **(_xii_)** & **(_yi_)** \\ \(\ast\) & 2013/1 & 1308 & & D \\ \(\ast\) & 2013/2 & 757 & & U \\ \(\ast\) & 2013/3 & 884 & & U \\ \(\ast\) & 2013/4 & 1062 & & F \\ \(\ast\) & 2013/5 & 1118 & & F \\ \(\ast\) & 2013/6 & 1097 & & U \\ \(\ast\) & 2013/7 & 1346 & & D \\ \(\ast\) & 2013/8 & 903 & & U \\ \(\ast\) & 2013/9 & 1312 & & F \\ \(\ast\) & 2013/10 & 1289 & & U \\ \(\ast\) & 2013/11 & 1424 & & U \\ \(\ast\) & 2013/12 & 1965 & & D \\ \(1\) & 2014/1 & 1437 & D & 1 & D \\ \(2\) & 2014/2 & 624 & U & 2 & U \\ \(3\) & 2014/3 & 1294 & U & 3 & F \\ \(4\) & 2014/4 & 1393 & F & 4 & F \\ \(5\) & 2014/5 & 1284 & F & 5 & U \\ \(6\) & 2014/6 & 1424 & U & 6 & F \\ \(7\) & 2014/7 & 1469 & D & 7 & D \\ \(8\) & 2014/8 & 1190 & U & 8 & U \\ \(9\) & 2014/9 & 1454 & F & 9 & U \\ \(10\) & 2014/10 & 1630 & U & 10 & F \\ \(11\) & 2014/11 & 1565 & U & 11 & U \\ \(12\) & 2014/12 & 2135 & D & 12 & F \\ \(13\) & 2015/1 & 2040 & D & 1 & D \\ \(24\) & 2015/12 & 1718 & F & 12 & U \\ \(25\) & 2016/1 & 2045 & D & 1 & D \\ \(71\) & 2019/11 & 1461 & F & 11 & U \\ \(72\) & 2019/12 & 1927 & F & 12 & D \\ \hline \end{tabular} \(\ast\) The records of 2013 are for ensuing predictor variables.  
\end{table} 
Table 4: The dataset of the classical model for BMW.