Table 1: Competition results point scores for various matches. Home team Away team Home team Away team BrisbanCanberra 67 52 82 98 68 49 94 79 41 81 Brisbane Brisban Adelaide AdelaidDarwin Canberra Adelaide Example equations Suppose that there are four teams in some par- ticular sports league: (2) Adelaide, (3) Brisbane, (4) Canberra, and (5) Darwin. Consider the competition results shown in Table 1- let denote the rating' of team i for i 2,3, 4, 5 corresponding to the listerd tesms i 1 is to be used soon). For these rating to be meaningful they must predict something: suppose the difference in ratings reflects the outcome of matches played between two teams. So, when the home-team rating is ri and the away-team rating is ry, then we aim for the difference TI-j-s-sy, where s is the home-team score and aj is the away-team score. For the competition results of Table 1, this gives the following five equations we would like to satisfy: r3-2 82-79- 3 2-98-4157, T4-2-68-81--13. Home ground advantage But a complication is that most sports recognise there is a home ground advantage. How do we factor in such an effect? The advantage (if any) is unknown, so let the advantage be denoted by the new unknow To this to be meaningful it must be reflected in the match reaults. Suppose that a home team is expected to score 1 more points than its ratings would predict. That is, the differences in the scores for each match shoukd be s-sj1) - rj when i is the home team. Upon accounting for home ground advantage the system (3.1) then becomes that the home ground advantage means (3.2) 1-13. This forms five equations for five unknowns: let's solve it! 3.1. Write (3.2) in the matrix-vector form Azb. Code the matrix and right-hand side into MATLAB and execute-Ab-what problem arises to stop us proceeding this way? What is the computed condition mimber of A? Table 1: Competition results point scores for various matches. Home team Away team Home team Away team BrisbanCanberra 67 52 82 98 68 49 94 79 41 81 Brisbane Brisban Adelaide AdelaidDarwin Canberra Adelaide Example equations Suppose that there are four teams in some par- ticular sports league: (2) Adelaide, (3) Brisbane, (4) Canberra, and (5) Darwin. Consider the competition results shown in Table 1- let denote the rating' of team i for i 2,3, 4, 5 corresponding to the listerd tesms i 1 is to be used soon). For these rating to be meaningful they must predict something: suppose the difference in ratings reflects the outcome of matches played between two teams. So, when the home-team rating is ri and the away-team rating is ry, then we aim for the difference TI-j-s-sy, where s is the home-team score and aj is the away-team score. For the competition results of Table 1, this gives the following five equations we would like to satisfy: r3-2 82-79- 3 2-98-4157, T4-2-68-81--13. Home ground advantage But a complication is that most sports recognise there is a home ground advantage. How do we factor in such an effect? The advantage (if any) is unknown, so let the advantage be denoted by the new unknow To this to be meaningful it must be reflected in the match reaults. Suppose that a home team is expected to score 1 more points than its ratings would predict. That is, the differences in the scores for each match shoukd be s-sj1) - rj when i is the home team. Upon accounting for home ground advantage the system (3.1) then becomes that the home ground advantage means (3.2) 1-13. This forms five equations for five unknowns: let's solve it! 3.1. Write (3.2) in the matrix-vector form Azb. Code the matrix and right-hand side into MATLAB and execute-Ab-what problem arises to stop us proceeding this way? What is the computed condition mimber of A
