## 3 New Offensive-Defensive Plus-Minus Ratings
### Example Segment  
To illustrate the model, a segment is selected from a match played between Barcelona and Athletic Bilbao in the Spanish Primera Division on 17 January 2016. The selected segment of the match started after a substitution in minute 6 and ended with a new substitution in minute 46. Before this, the goalkeeper of the away team had been shown a red card, and the home team had been awarded a penalty shot, which would be taken by [PERSON] in minute 7. The match was tied when the segment started. The following assumes that the ratings are being estimated on 21 June 2017, and the notation is simplified by dropping references to the match \(m\) and the segment \(s\). The duration of the selected segment is \(d=40\) min, and the home team scored twice and the away team zero times during these minutes. The rating model contains two quadratic terms tied directly to this segment, having, respectively, \(f^{H.RHS}=2\) and \(f^{A.RHS}=0\) as the constant terms, representing the goals scored by each team. The weights of the segment become \(w^{TIME}=0.867\), \(w^{DURATION}=1.133\), and \(w^{GOALS}=1\), resulting in \(w=0.983\). The number of non-zero linear terms of \(f^{HLHS}\) and \(f^{A.LHS}\), as appearing in the objective function terms (1) and (2), is 44 and 45, respectively. As there is one player sent off for the away team, the segment specific terms are \(f^{H.SEGMENT}=\beta_{1}^{D.ARED}\) and \(f^{ASEGMENT}=-\beta_{1}^{O.ARED}\). The match specific terms represent the home field advantage, with \(f^{H.MATCH}=\beta_{2}^{O.HFA}\) and \(f^{A.MATCH}=-\beta_{2}^{D.HFA}\), with the Spanish league represented by \(c^{A}=2\). As goalkeepers do not have offensive ratings, and since the away team has one player missing, the number of players contributing offensively is \(|P_{h}^{O}|=10\) and \(|P_{a}^{O}|=9\) for the two teams, while the number of players contributing defensively is \(|P_{h}^{D}|=11\) and \(|P_{a}^{D}|=10\). These numbers are used to normalize the sum of contributions from player ratings, \(f^{O.PLAYER}\) and \(f^{D.PLAYER}\), to \(f^{H.LHS}\) and \(f^{A.LHS}\). Table 1 shows the coefficients for the age variables in the selected segment, before scaling by \(d/90\) and multiplying by the segment weight \(w\). These coefficients arise from summing over the players appearing in the segment. Based on the coefficients, one can see that the home team has one player aged between 23 and 24, and two players aged between 31 and 32, excluding the goalkeeper: the sum of coefficients for offensive contributions related to age sum up to 1.1 for ages 23 and 24 and to 2.2 for ages 31 and 32, corresponding to the fact that these coefficients have been scaled by 11 and divided by the number of offensive players. The coefficients of the league components, \(\beta^{O.COMP}\) and \(\beta^{D.COMP}\) are given in Table 2. The players involved in the match had previously appeared in five different league competitions, most commonly being the Spanish Primera Division. The away team had six players previously appearing in the Spanish Segunda Division, each contributing with \(\frac{11}{|C_{p}||P_{a}^{D}|}\) and \(\frac{11}{|C_{p}||P_{a}^{D}|}\) to the given coefficients, with \(|C_{p}|=2\), as these players only appeared in two different league competitions. Finally, both \(f^{H,LHS}\) and \(f^{A,LHS}\) have, in total, twenty occurrences of \(\beta_{p}^{O}\) and \(\beta_{p}^{D}\), representing the main player rating components. With the proposed model, each match is, on average, divided into around 6.3 segments, each of which provides two squared expressions for the objective function. Less than half of the terms in these expression relate directly to player ratings, whereas the rest are used to estimate home field advantages, age effects, and league effects.