Comparison of Computerized Ranking Score Distributions

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LSQRANK
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Joined: Fri Nov 21, 2014 6:05 pm

Comparison of Computerized Ranking Score Distributions

Post by LSQRANK » Wed Jan 03, 2018 12:40 pm

Now that the Girls MN HS Hockey season is at about its mid-point, the various computerized ranking methods have finally accumulated enough data to start looking reasonable. For grins, I took the ranking results for the 3 local ranking methods (LSQRank, QRF & Krach) and compared their distributions. The LSQRank & QRF were rescaled using the range of the Krach score data so that we can visualize the three curves in the same chart.

Notice that the Krach trend line drops off dramatically within the first several teams, then flattens out which seems to indicate that the talent pool among the bottom 100 MN HS girls team is relatively homogenous. I'm not sure that this is the best of assumptions knowing the various levels of the girl's teams around the region. The Krach distribution can be modeled using the following exponential function

Y = AX^B where A = 1466.5, B = -1.0123 and X = Team ordinal rank

To make the Krach rating profile useful for the purposes of predicting actual future game score between teams, you will need to transform its rating profile using the following transformation equation:

Y = A*10^(Bx) where A = 0.00012417, B = 0.34586 and X = Krach Score

Here is a graph of the three distributions

Image

Here is the game result comparison for Tue Jan 02, 2018

Code: Select all

                                                                                                 &#91;<--------- LSQRANK ------------>&#93;
                                                             &#91;  RAW   &#93; &#91; Corrected &#93;            &#91;<-------- Jan 01 Profile ------>&#93;
YYYY MM DD   AWAY_TEAM                 HOME_TEAM              ATS  HTS   ATS    HTS    &#91; PIM  &#93;   AWAY         HOME        PREDICT   ACTUAL   Residual
---- -- --   -----------------------   ---------------------- ---  ---   ---    ---    --------   --------     --------    -------   ------   -------
2018 01 02   ChisagoLakes              Minneapolis              3    2   3.0    2.0     14   4    989.8354     991.8468    -2.0114   +1.00    -3.0114
2018 01 02   MankatoEast               NewUlm                   0    6   0.0    5.2     10   4    987.7293     992.6719    +4.9426   +5.20    -0.2574
2018 01 02   Blake                     Breck                    3    1   2.8    1.0      4   6    999.6263     996.6429    +2.9834   +1.80    +1.1834
2018 01 02   MinnehahaUnited           St.PaulUnited            1    10  0.8   11.75     6   6    989.8041     994.6318    +4.8277  +10.95    -6.1223
2018 01 02   Moorhead                  Bemidji                  4    0   4.0    0.0      6   2    991.1780     987.9631    +3.2149   +4.0     -0.7851
2018 01 02   Wayzata                   EdenPrairie              1    4   1.0    4.0      6   8    996.7271     996.9053    +0.1782   +3.0     -2.8218
2018 01 02   Hopkins/St.LouisPark      Minnetonka               1    6   1.0    5.4      6   4    994.8999     997.2163    +2.3164   +4.4     -2.0836
2018 01 02   Centennial                ChamplinPark/CoonRapids  6    3   5.6    3.25     8   8    999.6040     994.1359    +5.4681   +2.35    +3.1181
2018 01 02   Owatonna                  Austin                   7    1   7.0    1.0      2   8    991.1823     986.8917    +4.2906   +6.0     -1.7094
2018 01 02   Osseo/ParkCenter          MoundWestonka            0    2   0.0    2.0      2   0    990.2798     992.1155    +1.8357   +2.0     -0.1643
2018 01 02   RiverLakes                St.Cloud                 0    3   0.0    3.0      0   4    991.8692     993.6653    +1.7961   +3.0     -1.2039
2018 01 02   NorthernLakes             DetroitLakes             6    2   5.8    2.0      6   8    986.7534     987.4946    -0.7412   +3.8     -4.5412
2018 01 02   Hastings                  Stillwater               1    1   1.0    1.0      4   6    990.1077     992.9745    -2.8668    0.0     -2.8668
2018 01 02   Litchfield/Dassel-Cokato  HenrySibley              2    3   2.0    2.8      10  8    987.1747     990.2014    +3.0267   +0.8     +2.2267
2018 01 02   WhiteBearLake             Hill-Murray              0    3   0.0    2.8      6   6    995.7203     998.7587    +3.0384   +2.8     +0.2384
2018 01 02   RochesterMayo             RochesterCentury         -    -   -.-    -.-      -   -    991.6283     984.4581    +7.1702   ____     _______
2018 01 02   Eveleth-Gilbert           DuluthMarshall           1    6   0.8    6.05     4  12    988.5497     993.4131    +4.8634   +5.25    -0.3866
2018 01 02   EastGrandForks            FargoNorth/South         5    4   4.8    3.8      6   4    993.6224     994.0375    -0.4151   +1.00    -1.4151
2018 01 02   Viroqua                   RockCounty               0    3   0.0    3.0      -   -    987.9052     991.8922    +3.9870   +3.00    +0.9870
2018 01 02   AppletonUnited            USMWildcats              -    -   -.-    -.-      -   -    987.3229     991.1075    +3.7846   _____    _______
2018 01 02   CapCityCougars            Icebergs                 -    -   -.-    -.-      -   -    990.2725     985.2761    +4.9964   _____    _______
2018 01 02   GrandForks                DevilsLake               3    0   3.0    0.0      -   -    986.6668     984.4254    +2.2414   +3.00    -0.7586
2018 01 02   Mandan                    Jamestown                4    0   4.0    0.0      -   -    984.9303     983.1570    +1.7733   +4.00    -2.2267
                                                                                                                                              =======
																																	 RMSE =    2.42 
																																	  
LSQRANK Predicted winner proportion  16 of 20 games = 80.00%
QRF	    Predicted winner proportion  15 of 19 games = 78.95%
KRACH   Predicted winner proportion  13 of 16 games = 81.25%

MN Girls Rankings&#58; January 2, 2018

&#91;<------------------------------  LSQRANK  &#93;  &#91;<------------------------------------------ QRF &#93;  &#91;<--------------------------------- KRACH  &#93;          
==============================================================================================================================================
   1                        Edina 1000.0000    1	    Blake &#40;14-0-1&#41;	                   120.3     1   Blake                        1321.266 
   2                        Blake  999.6263    2	    Centennial &#40;13-1-1&#41;	               113.7     2   Centennial                   1111.892 
   3                   Centennial  999.6040    3	    Eagan &#40;14-2&#41;	                   110.7     3   Edina                        523.657  
   4                  Hill-Murray  998.7587    4	    Edina &#40;11-1-1&#41;	                   106.5     4   Eagan                        298.647  
   5                   Farmington  997.6146    5	    Farmington &#40;11-3&#41;	               100.7     5   Maple Grove                  234.991  
   6                      Andover  997.6012    6	    Andover &#40;10-3-1&#41;	                95.5     6   Andover                      220.735  
   7                       Blaine  997.5359    7	    Minnetonka &#40;9-3-3&#41;	                90.4     7   Blaine                       183.136  
   8                      Warroad  997.4886    8	    Forest Lake &#40;10-2-2&#41;	            89.7     8   Hill-Murray                  180.47   
   9                   MapleGrove  997.2404    9	    Blaine &#40;10-3-1&#41;	                    87.0     9   Brainerd/Little Falls        136.332  
  10                   Minnetonka  997.2163    10	    Hill-Murray &#40;12-3&#41;	                86.4     10  Minnetonka                   132.482  
  11                        Eagan  997.2013    11	    Brainerd/Little Falls &#40;13-3&#41;        83.9     11  Forest Lake                  130.817  
  12                  EdenPrairie  996.9053    T-12	    Lakeville South &#40;9-6&#41;	            83.8     12  Farmington                   129.695  
  13                      Wayzata  996.7271    T-12	    Roseau &#40;11-4&#41;	                    83.8     13  Roseau                       94.565   
  14               LakevilleSouth  996.6654    14	    Red Wing &#40;12-4&#41;	                    81.6     14  Eden Prairie                 93.244   
  15                        Breck  996.6429    15	    Eden Prairie &#40;10-6-1&#41;	            79.8     15  Warroad                      76.718   
  16                       Roseau  996.0683    16	    White Bear Lake &#40;12-4-1&#41;	        79.7     16  Lakeville South              76.204   
  17                   ForestLake  996.0018    17	    Maple Grove &#40;9-4-3&#41;	                77.6     17  White Bear Lake              72.79    
  18           ElkRiver/Zimmerman  995.8557    18	    Lakeville North &#40;9-6&#41;	            77.1     18  Thief River Falls            68.295   
  19               LakevilleNorth  995.7372    19	    Rochester Mayo &#40;11-3&#41;	            76.4     19  Breck                        67.37    
  20                WhiteBearLake  995.7203    20	    Wayzata &#40;8-5-1&#41;	                    76.1     20  Lakeville North              58.784   
  21                     Shakopee  995.7012    21	    South St. Paul &#40;12-3-1&#41;	            76.0     21  South St Paul                53.969   
  22            Cretin-DerhamHall  995.6613    22	    Duluth Marshall &#40;9-5&#41;	            73.1     22  Grand Rapids/Greenway        52.928   
  23         Brainerd/LittleFalls  995.6452    23	    North Wright County &#40;10-5-1&#41;        72.4     23  Wayzata                      51.68    
  24                  SouthStPaul  995.2755    24	    St. Cloud &#40;10-4-1&#41;	                72.3     24  Shakopee                     47.057   
  25                        Anoka  995.2191    25	    Holy Family Catholic/Waconia &#40;11-5&#41;	71.1     25  Elk River/Zimmerman          44.945   
  26           HolyFamily/Waconia  995.0869    26	    St. Paul United &#40;9-6&#41;	            70.7     26  Red Wing                     43.933   
  27                  AppleValley  995.0032    27	    Thief River Falls &#40;11-1-2&#41;          70.6     27  Cretin-Derham Hall           39.365   
  28         GrandRapids/Greenway  994.9490    28	    Warroad &#40;11-2-2&#41;	                69.8     28  Holy Family/Waconia          38.661   
  30         Hopkins/St.LouisPark  994.8999    29	    Hopkins/Park &#40;10-5&#41;	                69.7     29  St. Cloud                    37.307   
  31                     Eastview  994.8671    30	    Grand Rapids/Greenway &#40;11-4-1&#41;	    69.5     30  Hopkins/St. Louis Park       36.434   
  32                       Rogers  994.8450    31	    CambridgeIsanti/Mora/Pine C &#40;12-4&#41;	69.3     31  St. Paul United              36.372   
  33                St.PaulUnited  994.6318    32	    Shakopee &#40;8-7-1&#41;	                69.2     32  Anoka                        33.93    
  34                   Burnsville  994.5434    33	    Cretin-Derham Hall &#40;8-6-1&#41;	        69.0     33  Alexandria                   33.369   
  35                      RedWing  994.5353    34	    Breck &#40;6-7-1&#41;	                    67.5     34  Prior Lake                   30.86    
  36            NorthWrightCounty  994.4183    35	    Burnsville &#40;7-7-2&#41;	                65.9     35  Duluth Marshall              29.034   
  37                   Alexandria  994.2573    36	    Alexandria &#40;9-6&#41;	                65.8     36  North Wright County          27.165   
  39                   MoundsView  994.2224    37	    Champlin Park/Coon Rapids &#40;9-4-1&#41;	65.1     37  Rogers                       25.746   
  40      ChamplinPark/CoonRapids  994.1359    38	    Elk River/Zimmerman &#40;6-6-2&#41;	        65.0     38  Cambridge-Isanti/Mora/PC     25.715   
  42            Chaska/Chanhassen  993.8835    39	    Rogers &#40;7-5-3&#41;   	                64.9     39  East Grand Forks             25.388   
  43           Proctor/Hermantown  993.8751    40	    Northfield &#40;9-6-1&#41;	                64.5     40  Mounds View                  25.036   
  44        Benilde-St.Margaret's  993.8239    41	    St. Francis/North Branch &#40;9-5&#41;      64.3     41  Rochester Mayo               22.94    
  45                    PriorLake  993.8149    42	    Prior Lake &#40;7-7-1&#41;               	64.0     42  Burnsville                   22.729   
  46              ThiefRiverFalls  993.7694    43	    Anoka &#40;6-7-1&#41;	                    63.9     43  Proctor/Hermantown           20.466   
  47                     St.Cloud  993.6653    44	    Mounds View &#40;7-7-1&#41;	                61.4     44  Sartell/Sauk Rapids          19.252   
  48               EastGrandForks  993.6224    45	    Sartell/Sauk Rapids &#40;7-7-2&#41;	        60.7     45  Eastview                     18.931   
  49                  DodgeCounty  993.4684    46	    New Ulm &#40;12-3&#41;	                    60.3     46  Champlin Park/Coon Rapids    17.072   
  50               DuluthMarshall  993.4131    47	    Owatonna &#40;9-4&#41;	                    59.9     47  Apple Valley                 16.858   
  51                    NewPrague  993.2875    48	    East Grand Forks &#40;9-3-1&#41;	        59.0     48  St. Francis/North Branch     16.11    
  52     Cambridge-Isanti/Mora/PC  993.2645    49	    Delano/Rockford &#40;9-7&#41;	            58.2     49  Northfield                   15.994   
  53              Delano/Rockford  993.1977    50	    Apple Valley &#40;6-9-1&#41;	            57.9     50  Rochester Lourdes            15.217   
  54                    Roseville  993.0962    51	    Proctor/Hermantown &#40;8-8&#41;	        56.9     51  New Ulm                      15.16    
  55           ParkofCottageGrove  993.0781    52	    Dodge County &#40;10-5&#41;	                56.7     52  Chaska/Chanhassen            14.527   
  56                    EastRidge  992.9781    53	    Eastview &#40;4-9-2&#41;	                55.5     53  Benilde-St. Margaret's       14.025   
  58                   Stillwater  992.9745    54	    Mahtomedi &#40;8-6-1&#41;	                54.2     54  Dodge County                 13.998   
  59                        Orono  992.9730    55	    Hutchinson &#40;7-7-1&#41;	                53.4     55  Delano/Rockford              13.953   
  60             RochesterLourdes  992.8905    56	    Princeton/Big Lake/Becker &#40;9-7-2&#41;	53.2     56  Park of Cottage Grove        13.523   
  61       St.Francis/NorthBranch  992.8862    57	    Faribault &#40;9-6&#41;	                    52.7     57  Mahtomedi                    12.941   
  63             Hibbing/Chisholm  992.8453    58	    C-E-C &#40;5-7-1&#41;	                    52.1     58  New Prague                   11.3     
  65                    Mahtomedi  992.7326    59	    Rochester Lourdes/Dover-E &#40;10-8&#41;	51.8     59  Princeton                    10.723   
  66         Cloquet-Esko-Carlton  992.6720    60	    Park of Cottage Grove &#40;6-8-1&#41;	    51.5     60  East Ridge                   10.175   
  67                       NewUlm  992.6719    61	    Buffalo/Maple Lake &#40;6-11&#41;	        50.7     61  Cloquet-Esko-Carlton         9.315    
  68           Sartell/SaukRapids  992.5963    62	    Mankato East/Loyola &#40;7-6&#41;	        50.1     62  Fergus Falls                 9.243    
  70                  FergusFalls  992.4310    63	    East Ridge &#40;6-10&#41;	                49.8     63  Orono                        9.022    
  72                    Princeton  992.3541    64	    Mankato West &#40;10-7&#41;	                48.8     64  Hutchinson                   8.37     
  73                       Duluth  992.3305    65	    New Prague &#40;6-8-1&#41;	                48.3     65  Buffalo                      8.357    
  75         BloomingtonJefferson  992.2670    T-66	    Chaska/Chanhassen &#40;5-9-1&#41;	        47.1     66  Owatonna                     8.354    
  76               SpringLakePark  992.2527    T-66	    Spring Lake Park &#40;7-9&#41;	            47.1     67  Bloomington Jefferson        7.113    
  77                   Northfield  992.1600    68	    Orono &#40;6-10&#41;	                    46.6     68  Duluth                       7.044    
  78                      Buffalo  992.1525    69	    Bloomington Jefferson &#40;6-9&#41; 	    46.3     69  Stillwater                   7.034    
  79                MoundWestonka  992.1155    70	    Fergus Falls &#40;9-8&#41;	                45.7     70  River Lakes                  6.982    
  80                   Hutchinson  991.9263    71	    Benilde-St. Margaret's &#40;5-9-1&#41;	    45.5     71  Moorhead                     6.901    
  82                   RiverLakes  991.8692    72	    River Lakes &#40;6-9&#41;	                45.4     72  Faribault                    6.826    
  83                  Minneapolis  991.8468    73	    Henry Sibley &#40;5-8&#41;	                43.3     73  Hibbing/Chisholm             6.667    
  86                RochesterMayo  991.6283    74	    Hibbing/Chisholm &#40;6-9-2&#41;	        43.2     74  Roseville                    6.307    
  88                      Willmar  991.2216    75	    Totino-Grace &#40;6-8-1&#41;	            43.0     75  Spring Lake Park             5.779    
  89                     Owatonna  991.1823    76	    Roseville &#40;4-10-1&#41;	                42.9     76  Moose Lake Area              5.606    
  90                     Moorhead  991.1780    77	    Mound Westonka &#40;5-7-2&#41;	            42.2     77  Willmar                      5.544    
  95           InternationalFalls  990.8837    T-78	    Willmar &#40;6-9&#41;	                    42.0     78  Mound Westonka               5.238    
  96                 Totino-Grace  990.8821    T-78	    Luverne &#40;11-5-2&#41;	                42.0     79  International Falls          5.053    
  97                MooseLakeArea  990.8022    80	    Minnehaha &#40;8-6&#41;	                    41.0     80  Minneapolis                  4.972    
  98                   HolyAngels  990.7048    81	    International Falls &#40;5-8-1&#41;	        40.7     81  Totino-Grace                 4.398    
 104                       Simley  990.3408    82	    Moose Lake Area &#40;5-7-1&#41;	            40.5     82  Osseo/Park Center            3.109    
 105          NorthSt.Paul/Tartan  990.3246    83	    Stillwater Area &#40;3-9-2&#41;	            40.1     83  Mankato East                 3.037    
 106             Osseo/ParkCenter  990.2798    84	    Minneapolis &#40;6-9&#41;	                39.8     84  Mankato West                 2.67     
 109                  HenrySibley  990.2014    85	    Austin &#40;5-8&#41;	                    39.5     85  Henry Sibley                 2.389    
 110                    Faribault  990.2005    86	    Duluth &#40;4-11-2&#41;	                    39.2     86  Holy Angels                  2.346    
 113                     Hastings  990.1077    87	    Marshall &#40;7-8&#41;	                    38.8     87  Chisago Lakes                2.024    
 121                 ChisagoLakes  989.8354    88	    Moorhead &#40;3-9&#41;	                    38.5     88  Rosemount                    1.956    
 122              MinnehahaUnited  989.8041    89	    Osseo/Park Center &#40;5-8-1&#41;	        38.4     89  Simley                       1.934    
 134                    Rosemount  989.2863    90	    Simley &#40;4-9-1&#41;	                    37.0     90  Eveleth-Gilbert              1.784    
 136  RobbinsdaleArmstrong/Cooper  989.1184    91	    Rosemount &#40;5-9-1&#41;	                36.3     91  Luverne                      1.776    
 137                     Irondale  989.0724    92	    Prairie Centre &#40;7-7&#41;	            34.6     92  North St. Paul/Tartan        1.727    
 140                PrairieCentre  988.9120    93	    Chisago Lakes Area &#40;4-11&#41;	        33.3     93  Prairie Centre               1.692    
 142                   NorthShore  988.8156    94	    Windom Area &#40;6-6-1&#41;	                33.2     94  Minnehaha United             1.633    
 147              Eveleth-Gilbert  988.5497    T-95	    Holy Angels/Richfield &#40;4-9-1&#41;	    32.2     95  Marshall                     1.526    
 150                  MankatoWest  988.4001    T-95	    Rochester John Marshall &#40;5-11&#41;	    32.2     96  Irondale                     1.509    
 151                     Marshall  988.3303    97	    North Shore &#40;5-10&#41;	                30.9     97  Hastings                     1.434    
 153                     Woodbury  988.2157    98	    Detroit Lakes &#40;6-9&#41;	                30.6     98  Rochester John Marshall      1.353    
 160                      Bemidji  987.9631    99	    Eveleth-Gilbert &#40;6-7-1&#41;	            30.4     99  Detroit Lakes                1.335    
 166                  MankatoEast  987.7293    T-100	Minnesota River &#40;5-9&#41;	            30.2     100 Austin                       1.234    
 171                 DetroitLakes  987.4946    T-100	North/Tartan &#40;4-9-3&#41;	            30.2     101 Northern Lakes               1.119    
 174                      Luverne  987.3180    102	    Northern Lakes &#40;5-7&#41;	            29.7     102 North Shore                  1.019    
 175     Litchfield/Dassel-Cokato  987.1747    103	    Hastings &#40;2-13&#41;	                    29.4     103 Bemidji                      1.001    
 180                       Austin  986.8917    104	    Waseca &#40;4-10&#41;	                    28.8     104 Minnesota River              0.899    
 188                NorthernLakes  986.7534    105	    Irondale/St. Anthony &#40;3-13&#41;	        27.6     105 Windom                       0.771    
 190        RochesterJohnMarshall  986.5773    106	    Albert Lea &#40;3-11&#41;	                24.5     106 Litchfield/Dassel-Cokato     0.722    
 194               MinnesotaRiver  986.3115    107	    Litchfield/Dassel-Cokato &#40;2-12&#41;	    24.3     107 Woodbury                     0.668    
 198                       Waseca  986.2136    108	    Rochester Century &#40;3-11&#41;	        23.9     108 Robbinsdale Armstrong/Cooper 0.53     
 203                       Windom  985.6212    109	    Bemidji &#40;2-13&#41;	                    23.7     109 Waseca                       0.44     
 209                St.PaulBlades  985.1118    110	    Worthington/Fulda &#40;3-9&#41;	            23.5     110 Albert Lea                   0.416    
 211                    AlbertLea  985.0718    111	    Robbinsdale Armstrong/Cooper &#40;1-13&#41;	21.9     111 Rochester Century            0.362    
 224             RochesterCentury  984.4581    112	    Woodbury Area &#40;2-13&#41;	            21.5     112 Worthington                  0.312    
 232                   ParkRapids  984.1031    113	    St. Paul Blades &#40;1-15&#41;	            17.0     113 St. Paul Blades              0.218    
 243                    Crookston  983.1534    114	    Park Rapids Area &#40;2-13&#41;	            16.5     114 Park Rapids                  0.143    
 251                  Worthington  982.1207    115	    Crookston &#40;2-12&#41;	                15.6     115 Crookston                    0.129    
 272        MorrisArea/BensonArea  980.2654    116	    Fairmont &#40;1-14&#41;	                    13.9     116 Fairmont                     0.071    
 274                     Fairmont  979.8558    117	    Morris/Benson Area &#40;1-9&#41;	        11.8     117 Morris Area/Benson Area      0.053    
																																  		   
QRF ND Girls Rankings&#58; January 1, 2018         LSQRANK
======================================         ================================
1	Bismarck &#40;7-3&#41;	            79.4           41   FargoNorth/South   994.0375 
2	West Fargo &#40;8-3-1&#41;	        77.5           64           Bismarck   992.7816 
3	Fargo North/South &#40;9-2&#41;	    73.5           92    WestFargoUnited   990.9773 
T-4	Fargo Davies &#40;6-4&#41;	        57.1          103        FargoDavies   990.3419 
T-4	Minot &#40;5-5&#41;	                57.1          158              Minot   988.0195 
6	Grand Forks &#40;5-4&#41;	        37.6          189         GrandForks   986.6668 
7	Devils Lake &#40;3-6&#41;	        32.9          212          Dickinson   984.9304 
8	Dickinson &#40;2-4&#41;	            25.4          213             Mandan   984.9303 
9	Mandan &#40;2-9&#41;	            25.3          225         DevilsLake   984.4254 
10	Williston &#40;3-6&#41;	            21.9          240          Williston   983.2816 
11	Jamestown &#40;2-8&#41;	             9.8          242          Jamestown   983.1570 
                                              247        Richland19U   982.9304 
                                              277     WatfordCity19U   979.2816 

ghshockeyfan
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KRACH background info

Post by ghshockeyfan » Wed Jan 03, 2018 7:18 pm

https://www.collegehockeynews.com/ratings/krach.php

Some background info on the KRACH approach and links to other related info.

Very interesting comparison above!

LSQRANK
Posts: 415
Joined: Fri Nov 21, 2014 6:05 pm

Post by LSQRANK » Wed Jan 03, 2018 11:02 pm

After spending a little bit more time thinking about how to transform the KRACH distribution into a profile that is useful in prediction of game score differentials, I came up with the following. There are two ways to approach this and below I have supplied the two models: (1) KRACH piece-wise transformation and (2) KRACH polynomial.

The polynomial produces a generalized approximation of a score value given an ordinal rank value. A more tailored method to use is the piece-wise transformation. Use this approach if you want a better approximation of transformed score values each time a new rating profile is updated. The basic equation is Y = A x Ln( B x KRACH_SCORE ) . There are one set of A&B parameters for KRACH scores between 0 and 130; and another set of A&B parameters for KRACH score above 130.

For example:
Team #1 K-score = 1321.266 ====> 20.29
Team #3 K-score = 523.657 ====> 19.11

Use A= 1.2779 & B = 5945.2

Therefore if Team #1 played Team #3, the expected game score differential would be 20.29 - 19.11 = 1.18 goals

Below is a graph of the transformed KRACH values and the modeled polynomial.

Image

Below are list of transformed and modeled values for both methods:

Code: Select all

KRACH &#58; Piece-wise Transformation
=================================
Y = A*Ln&#40;BX&#41;

Y     = Transformed KRACH Score used for Game Score Differential Prediction
X     = KRACH Rating Score
Ln&#40; &#41; = Natural Logarithm function
A & B are Transformation parameters
---------------------------------------------------------
A & B parameter values for KRACH scores between 0 and 130

A =  1.9987
B = 48.13
---------------------------------------------------------
A & B parameter values for KRACH scores above 130   

A =    1.2779 
B = 5945.2
=============================================================================
   
KRACH Polynomial
Y = A + BX + CX^2 + DX^3 + EX^4 + FX^5

A = +20.400      &#91;+/- 0.16684    &#93;
B = - 0.43119    &#91;+/- 0.028143   &#93;
C = + 0.014956   &#91;+/- 0.0014623  &#93;
D = -0.00029387  &#91;+/- 3.1270E-005&#93;
E = +2.6767E-006 &#91;+/- 2.9164E-007&#93;
F = -9.2576E-009 &#91;+/- 9.8376E-010&#93;


KRACH-Jan02   KRACH
Transformed   Polynomial     Rank
-----------   -------------  ---
20.29008602   19.9831147888   1  
20.06959958   19.5947777798   2  
19.10730434   19.2329583482   3  
18.38962321   18.8960502804   4  
18.08327216   18.5825082703   5  
18.00329064   18.2908468084   6  
17.76464853   18.0196390708   7  
17.74590761   17.7675158087   8  
17.38747067   17.533164237    9  
17.35086128   17.3153269238   10 
17.33469817   17.112800679    11 
17.26682163   16.9244354439   12 
16.83543607   16.7491331802   13 
16.80731893   16.5858467586   14 
16.41740583   16.4335788486   15 
16.40396976   16.2913808069   16 
16.3123586    16.1583515672   17 
16.18495742   16.0336365287   18 
16.15770166   15.9164264455   19 
15.88521868   15.8059563155   20 
15.71440997   15.7015042697   21 
15.67548068   15.602390461    22 
15.62778842   15.5079759538   23 
15.44048779   15.4176616124   24 
15.34870726   15.3308869908   25 
15.30318932   15.2471292211   26 
15.08375474   15.1659019031   27 
15.04768669   15.0867539933   28 
14.97643231   15.0092686937   29 
14.92910595   14.9330623413   30 
14.92570185   14.8577832968   31 
14.78679286   14.7831108341   32 
14.75347002   14.708754029    33 
14.59723868   14.6344506486   34 
14.4753314    14.5599660399   35 
14.34234154   14.4850920197   36 
14.23511108   14.409645763    37 
14.23270305   14.3334686923   38 
14.20712397   14.2564253667   39 
14.17921846   14.1784023711   40 
14.00446677   14.0993072051   41 
13.98599784   14.0190671721   42 
13.77638084   13.9376282687   43 
13.65416029   13.8549540734   44 
13.62055383   13.7710246358   45 
13.41396601   13.6858353658   46 
13.38875366   13.5993959227   47 
13.29804241   13.511729104    48 
13.28359872   13.4228697349   49 
13.1840626    13.3328635572   50 
13.17656179   13.2417661182   51 
13.09131445   13.1496416601   52 
13.0210251    13.056562009    53 
13.01717362   12.9626054639   54 
13.01073796   12.8678556857   55 
12.94817346   12.7724005868   56 
12.8602479    12.6763312194   57 
12.5892285    12.5797406652   58 
12.48447312   12.4827229245   59 
12.37962684   12.3853718046   60 
12.20312618   12.2877798099   61 
12.18761728   12.1900370302   62 
12.1392478    12.09223003     63 
11.98932101   11.9944407379   64 
11.98621428   11.8967453351   65 
11.98549666   11.7992131452   66 
11.66407298   11.7019055226   67 
11.64458984   11.6048747421   68 
11.64175038   11.5081628877   69 
11.62691977   11.4118007418   70 
11.60359678   11.3158066744   71 
11.58175603   11.220185532    72 
11.53464894   11.1249275265   73 
11.42370139   11.030007125    74 
11.2489561    10.9353819381   75 
11.18820921   10.8409916094   76 
11.16598128   10.7467567047   77 
11.05250206   10.6525776006   78 
10.98063357   10.558333374    79 
10.94833467   10.4638806913   80 
10.70314958   10.369052697    81 
10.00990296   10.2736579032   82 
9.96307153    10.1774790785   83 
9.705655532   10.0802721372   84 
9.483392889   9.98176502823   85 
9.447090264   9.88165662445   86 
9.152010194   9.77961561158   87 
9.08370626    9.67527937736   88 
9.061098616   9.5682529006    89 
8.899738842   9.45810764031   90 
8.890755906   9.34438042477   91 
8.834836586   9.22657234061   92 
8.793914127   9.10414762191   93 
8.722975373   8.97653253931   94 
8.58752571    8.84311428906   95 
8.565134767   8.70323988213   96 
8.463242166   8.55621503331   97 
8.347030966   8.40130305028   98 
8.320262262   8.23772372271   99 
8.163023801   8.06465221135   100
7.967499982   7.88121793712   101
7.780394329   7.6865034702    102
7.744772991   7.47954341911   103
7.529969214   7.2593233198    104
7.222979565   7.02477852476   105
7.091738458   6.77479309211   106
6.936365586   6.50819867464   107
6.473844086   6.22377340896   108
6.10188146    5.92024080456   109
5.989775443   5.59626863291   110
5.711874099   5.25046781654   111
5.414785285   4.88139131814   112
4.698235095   4.48753302963   113
3.855482376   4.06732666128   114
3.649551866   3.61914463078   115
2.456063083   3.14129695233   116
1.871667261   2.63203012573   117

ShakestheClown
Posts: 124
Joined: Tue Mar 03, 2009 4:11 pm

Post by ShakestheClown » Thu Jan 04, 2018 8:03 am

This is not intended to be a knock on your methodology, but I have no idea what I just read. :shock:

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