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This problem set consists of actual data played in Nov & Dec of 2021 in the MN State HS league.
Can you guess who the 6 teams actually are? (see below)
Supose that there exists six teams (A,B,C,D,E,F)
that play 12 games between each other, such that,
game #1 Team_A Team_B 4 3 ; Team_A beats Team_B by 1 goal
game #2 Team_B Team_C 4 2 ; Team_B beats Team_C by 2 goals
game #3 Team_C Team_D 4 1 ; Team_C beats Team_D by 3 goals
game #4 Team_D Team_A 6 0 ; Team_D beats Team_A by 6 goals
game #5 Team_E Team_B 3 5 ; Team_B beat Team_E by 2 goals
game #6 Team_E Team_D 5 3 ; Team_E beats Team_D by 2 goals
game #7 Team_E Team_C 5 6 ; Team_C beats Team_E by 1 goal
game #8 Team_F Team_B 1 3 ; Team_B beats Team_F by 2 goals
game #9 Team_F Team_D 2 3 ; Team_D beats Team_F by 1 goal
game #10 Team_F Team_A 3 2 ; Team_F beats Team_A by 1 goal
game #11 Team_F Team_C 0 3 ; Team_C beats Team_F by 3 goals
game #12 Team_A Team_B 5 4 ; Team_A beats Team_B by 1 goal
Let the ratings value for Team_A = 1000.000
Relative to Team_A,
calculate the rating values for the 5 other teams (B,C,D,E,F)
such that the sum of the square of the observation residuals are minimized.
If one chose the following rating values,
Team_A = 1000 (given)
Team_B = 1003
Team_C = 1003.5
Team_D = 1002
Team_E = 1002.5
Team_F = 1001
they would be close. What are the residual values in this case?
The residual value is denoted by little "v", and is calculated as
v = Observed_Value minus Predicted_Value
Game# H - A = Observed Away - Home = Predicted v v_squared
===== = = ======== ==== ==== ========= === =========
1 4 3 1 1000 1003 -3.0 4.0 16.00
2 4 2 2 1003 1003.5 -0.5 2.5 6.25
3 4 1 3 1003.5 1002 1.5 1.5 2.25
4 6 0 6 1002 1000 1.0 5.0 25.00
5 3 5 -2 1002.5 1003 -0.5 -1.5 2.25
6 5 3 2 1002.5 1002 0.5 1.5 2.25
7 5 6 -1 1002.5 1003.5 -1.0 0.0 0.00
8 1 3 -2 1001 1003 -2.0 0.0 0.00
9 2 3 -1 1001 1002 -1.0 0.0 0.00
10 3 2 1 1001 1000 1.0 0.0 0.00
11 0 3 -3 1001 1003.5 -2.5 -0.5 0.25
12 5 4 1 1000 1003 -3.0 4.0 16.00
_____
sum = 70.25
Using numerical matrix methods, one can solve for the minimized sum of the square of the residuals,
,i.e. the term "least squares solution" gets its name.
The actual least squares solution to this particular problem set is:
LSQRANK State Tier Rating Score Team
------------------------------------------------------------------------------
1 MN HS 1003.3750 Team_C MSHSL 0 xx 4 ( 3- 1- 0)
2 MN HS 1002.9375 Team_B MSHSL 0 xx 5 ( 3- 2- 0)
3 MN HS 1002.5000 Team_E MSHSL 0 xx 3 ( 1- 2- 0)
4 MN HS 1002.1875 Team_D MSHSL 0 xx 4 ( 2- 2- 0)
5 MN HS 1000.8750 Team_F MSHSL 0 xx 4 ( 1- 3- 0)
6 MN HS 1000.0000 Team_A MSHSL 0 xx 4 ( 2- 2- 0)
Avg V = 1.302
max V = 3.938
Snot = 3.174
DOFs = 6 ( equals 12 games - 6 teams)
To intreprete the LSQRank results:
If Team_D were to play Team_A in the future,
the best predicted outcome would be that Team_D would beat Team_A by approximately 1 goal.
contrasting the least squares solution to the KRACH Algorithm results:
Rank [--Team-] State League KRACH_Rating SOS WL_Ratio (WW-LL-TT)
---- ------------------------ ------------ ------ ---------------- ----
1 Team_C MN MSHSL 208.29 69.43 3.00 ( 3- 1- 0)
2 Team_B MN MSHSL 118.96 79.31 1.50 ( 3- 2- 0)
3 Team_A MN MSHSL 79.14 79.14 1.00 ( 2- 2- 0)
4 Team_D MN MSHSL 76.13 76.13 1.00 ( 2- 2- 0)
5 Team_E MN MSHSL 60.03 120.06 0.50 ( 1- 2- 0)
6 Team_F MN MSHSL 35.35 106.04 0.33 ( 1- 3- 0)
To intreprete the KRACH results:
If Team_D were to play a series of game against Team_A in the future,
the best predicted outcome would be that each team would win 50% of the games,
i.e. approximately an even split.
As of Dec. 24th, 2021 then latest profile run contained the following:
2052 games played by 496 teams = 1556 DOF's (degrees of freedom)
Break-down of teams included in the solution:
League #teams #classes
========== ====== ============
MN varsity 148 2 (A & AA )
MI varsity 135 3 (D1, D2, D3)
WI varsity 85 2 (D1 & D2)
ND varsity 20 1
SD varsity 11 1
MT varsity 8 1
MN JrGold 16
ND JrGold 9
MI JVPrep 41
Other 23 ; Other teams connected in the network
answer to Team ID's ( A=STA, B=CDH, C=Hermantown, D=HILL, E=BSM & F=Mahtomedi)