## 3 Results  
In this section, we apply the proposed ranking method to the data of NBA collected from the 2009-2010 season to 2018-2019 season. Five continuous variables (points, rebounds, assists, turnovers, and fouls) and one binary variable (home field advantage) are used as the input for the link function of the model. For each season, the game graph is constructed by the first half of the games (615 regular season games), and the total wins of the half are employed as the prior distribution for each team's strength. The number of games a team wins throughout the season is used as a measurement of the team's true strength. In the training process, the rating vector and the loss function are computed for the first six seasons, and the parameters are estimated by minimizing the total loss. Then, the testing process is done over the four seasons left. The estimated parameters in the link function \(\rho(X;\theta)\) are shown in Table 2. The estimated result is in line with common sense. These indicators of points, rebounds, and assists all have a positive effect on the team's strength; at the same time, turnovers and fouls have the opposite effect. In addition, home advantage will bring a \(\frac{1}{0.8266}\), 120.97% improvement to the home team's performance, which is similar to the ratio of the home team's winning percentage of 50%. The testing results are evaluated by three criteria: Kendall rank correlation, prediction accuracy, and playoff accuracy. For each season in the testing sets, the Kendall rank correlation is calculated between the rating vector estimated from the first 615 games and the true strength vector. The winning team of one game is predicted as the team with a higher rating value. Then, the prediction accuracy is computed for the 615 games left. A team ranked in the top eight in the conference will be predicted as a playoff team. Playoff accuracy is defined as the correct proportion of teams that are predicted to enter the playoffs. Comparisons are done between the proposed model and several popular ranking methods with applications primarily to sports [21], including the [PERSON] method, the [PERSON] method, the PageRank method, and the Elo method. The simplest model in which the team's rating is computed by the current wins is set as the baseline. The results are shown in Tables 3-5. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & **2015** & **2016** & **2017** & **2018** & **Average** \\ \hline Baseline & 0.6504 & 0.5937 & 0.6276 & 0.6146 & 0.6216 \\ Colley & 0.6667 & 0.6136 & 0.6488 & 0.6448 & 0.6445 \\ Massey & 0.6683 & 0.6153 & 0.6667 & 0.6439 & 0.6485 \\ PageRank & 0.6585 & 0.6036 & 0.6520 & 0.6244 & 0.6347 \\ Elo & 0.6715 & 0.6252 & 0.6504 & 0.6569 & 0.6510 \\ TeamRank & 0.6764 & 0.6169 & 0.6699 & 0.6504 & 0.6534 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Prediction accuracy of the results. Red values indicate the highest prediction accuracy for each season. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Points** & **Rebounds** & **Assists** & **Tumowers** & **Fouls** & **Home Advantage** \\ \hline \(\theta\) & 0.3303 & 0.0394 & 0.2938 & \(-\)0.1583 & \(-\)0.0292 & \(-\)0.1904 \\ \(\exp(\theta)\) & 1.3914 & 1.0402 & 1.3415 & 0.8536 & 0.9712 & 0.8266 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter estimation result. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & **2015** & **2016** & **2017** & **2018** & **Average** \\ \hline Baseline & 0.7823 & 0.7482 & 0.6343 & 0.7618 & 0.7317 \\ Colley & 0.7796 & 0.7959 & 0.6242 & 0.7140 & 0.7284 \\ Massey & 0.8029 & 0.7681 & 0.6706 & 0.6396 & 0.7203 \\ PageRank & 0.6302 & 0.6242 & 0.5035 & 0.5652 & 0.5807 \\ Elo & 0.7562 & 0.8005 & 0.6010 & 0.6721 & 0.7075 \\ TeamRank & 0.8309 & 0.8191 & 0.6891 & 0.6768 & 0.7540 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Kendall rank correlation of the results. Red values indicate the highest Kendall rank correlation for each season.