# Predicting Football Match Results Using a Poisson Regression Model
## 4 Analysis of the Results
### Experimental Results  
In Equations (4) and (5), the value of \(\mu\) is calculated as the logarithm of the average goals scored in away scores, which is equal to 0.199. The constant for all teams' home advantage \(\mu_{Home}\) is calculated as the difference of the logarithm of the home and away average goals scored, whose value is 0.290, quoted in Table 4.  
So, a typical away team will score 1.22 goals, while, on average, a home team is expected to score \(\exp(0.290)=133\%\) of the goals scored from the away opposition. This result coincides with the observed values, since 1.63 (average home goals scored) is 133% of 1.22. For the attacking and defensive quality for each team, we will need to calculate the average goals for and the average goals against each team for all the matches played. Table 5 presents those averages along with the attacking and defensive quality for each team. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Parameter** & **Value** \\ \hline \(\mu\) & 0.199 \\ \hline \(\mu_{Home}\) & 0.290 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results for parameters. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameter** & **Value** \\ \hline \(\chi^{2}\) & 20.37 \\ \hline Degrees of freedom & 16 \\ \hline \(p\)-value & 0.2039 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: \(\chi^{2}\) test results. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Team** & **Avg Goals for** &
\begin{tabular}{c} **Avg Goals** \\ **against** \\ \end{tabular} & **att Parameter** & **def Parameter** \\ \hline Manchester City & 2.47 & 0.87 & 0.707 & \(-0.34\) \\ \hline Arsenal & 2.32 & 1.13 & 0.641 & \(-0.075\) \\ \hline Manchester United & 1.53 & 1.13 & 0.224 & \(-0.075\) \\ \hline Newcastle & 1.79 & 0.87 & 0.383 & \(-0.34\) \\ \hline Liverpool & 1.97 & 1.24 & 0.481 & 0.014 \\ \hline Brighton & 1.89 & 1.39 & 0.44 & 0.134 \\ \hline Aston Villa & 1.34 & 1.21 & 0.095 & \(-0.008\) \\ \hline Tottenham Hotspur & 1.84 & 1.66 & 0.412 & 0.307 \\ \hline Brentford & 1.53 & 1.21 & 0.224 & \(-0.008\) \\ \hline Fulham & 1.45 & 1.39 & 0.171 & 0.134 \\ \hline Crystal Palace & 1.05 & 1.29 & \(-0.148\) & 0.055 \\ \hline Chelsea & 1.00 & 1.24 & \(-0.199\) & 0.014 \\ \hline Wolves & 0.82 & 1.53 & \(-0.402\) & 0.224 \\ \hline West Ham & 1.11 & 1.45 & \(-0.099\) & 0.171 \\ \hline Bournemouth & 0.97 & 1.87 & \(-0.226\) & 0.426 \\ \hline Nottingham Forest & 1.00 & 1.79 & \(-0.199\) & 0.383 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Attacking (att) and defensive (def) parameters. A more conservative approach for the calculation of the average goals for and against each team, from which the attacking and defensive parameters are measured, would be to cap match results where, for example, more than 5 goals were scored by a team to 5 (i.e., if a match ended 6-1 to be counted as 5-1). However, following the fact that the number of matches where more than 5 goals were scored by a team is very low (8 in total, and 6 of those refer to 6 goals by a team), no such management was applied as having no material impact on the results. An indicative example for the calculation of the expected number of goals for each team is presented in Equations (6) and (7). The match selected for the demonstration is Manchester City vs.Liverpool, which was played on 1 April 2023, fixture 29. MC refers to Manchester City and LIV to Liverpool, while xG refers to the expected number of goals that will be scored, utilizing Equations (4) and (5). \[xGMC=e^{h}\times e^{h_{Home}}\times e^{\text{atHMC}}\times e^{\text{def}_{ \text{LIV}}} \implies\] \[xGMC=e^{0.199}\times e^{0.290}\times e^{0.707}\times e^{0.014} \implies \tag{6}\] \[\textbf{xGMC}=\textbf{3.4}\]  
\[xGLIV=e^{h}\times e^{\text{atHUV}}\times e^{\text{def}_{\text{MC}}} \implies\] \[xGLIV=e^{0.199}\times e^{0.481}\times e^{-0.340} \implies \tag{7}\] \[\textbf{xGLIV}=\textbf{1.4}.\]  
The above means that, if Manchester City played Liverpool at home many times, Manchester City would on average score 3.4 goals, while Liverpool would score 1.4 goals.