# Predicting Football Match Results Using a Poisson Regression Model
## 4 Analysis of the Results
### Experimental Results
The observed result of the match was Manchester City vs. Liverpool 4-1. Furthermore, for this particular match, the probability of the final score being \(k-h\) (meaning that Manchester City scores k goals and Liverpool scores h goals), follows a double Poisson distribution and is calculated as follows:  
\[\frac{xGMC^{k}e^{-xGMC}}{k!}\ \times\frac{xGLIV^{h}e^{-xGLIV}}{h!} \tag{8}\]  
Table 6 presents the probabilities for all scores between the aforementioned teams of the above example using the above formula--the number of goals for home and away matches for each team, which has been capped at nine. For example, the home team--Manchester City--has a probability of 22.0% to score exactly three goals, while the away team--Liverpool--has a probability of 34.5% to score exactly one goal. As a result the probability of the final score being 3-1 is 22.0% \(\times\) 34.5% = 7.6% (the value in bold in Table 6). The score of 3-1 is also the one with the highest probability of 7.6%, followed by 6.8% for a score of 2-1 and 6.3% for a score of 4-1 (4-1 was the actual result of the match). Using Table 6, we can also calculate the probability of home win, draw, and away win by taking the sum of the probabilities of that particular outcome. Table 7 illustrates the corresponding probabilities. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Team** & **Avg Goals for** &
\begin{tabular}{c} **Avg Goals** \\ **against** \\ \end{tabular} & **att Parameter** & **def Parameter** \\ \hline Everton & 0.89 & 1.50 & \(-\)0.31 & 0.207 \\ \hline Leicester City & 1.34 & 1.79 & 0.095 & 0.383 \\ \hline Leeds & 1.26 & 2.05 & 0.035 & 0.52 \\ \hline Southampton & 0.95 & 1.92 & \(-\)0.253 & 0.454 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: _Cont._We observed that the probability of a home team win is considerably higher than that of a draw and an away team win, which is in alignment with the fact that Manchester City has the best attacking quality of the league. In order to test the efficacy of the model, two initial tests were conducted, using all available matches of the season (total of 380) for the parameters estimation present in the model, as per the methodology presented in the previous section. First, a random 20% sub-sample--i.e., 76 random matches--out of the total of 380 of the 2022-2023 Premier League's season were randomly selected, and the model estimates were compared against the observed results. As such, a sample of 76 random matches was created, and for each one, the model estimates were assessed and the difference between the model estimate and observed home and away goals was calculated. It should be noted that the difference between the model estimate and the observed value has been rounded to the nearest integer. The distribution of differences between model estimates and the observed result for home and away goals for all 76 matches included in the random sub-sample are available in the following tables for home and away scores. As the tables below suggest, the differences range mostly from \(-2\) (meaning the model estimated 2 less goals than actually scored) and 2 (the model estimated 2 goals more than the observed) for both home and away scores; however, a few cases in \(-3\) and 3 were recorded, which mostly refer to unexpected results with a strong favorite side. For the home scores, 75% of the differences are between \(-1\) and 1, while, in some cases, the model estimated 2 more goals than the actual scores (\(\approx\)18%). Furthermore, for away scores, the percentage of differences between \(-1\) and 1 rises to more than 90%. That means that most of the scores predicted by the model are either equal to the observed or to a deviation of just one goal. Those results, which are shown in Tables 8 and 9, are a strong indication that the selected model and methodology are robust and can accurately estimate the goals that a team is likely to score in a match.