paper: Weeks 1-2 Elo Analysis

[connor.worley]

Thread created automatically to discuss a document in CD-Media.

Weeks 1-2 Elo Analysis (http://www.chiefdelphi.com/media/papers/2972?) by connor.worley

Here is the Python code used to calculate the ratings:
import csv

teams = {}

def getElo(team):
if team not in teams:
teams[team] = 1200
return teams[team]

def setElo(team, elo):
teams[team] = elo

def calculateElo(fileName, kFactor):
reader = csv.reader(open(fileName, 'r'))

for row in reader:
if row[3] == "Q" or row[3] == "E":
averageRedElo = (getElo(row[9]) + getElo(row[10]) + getElo(row[11]))/3
averageBlueElo = (getElo(row[12]) + getElo(row[13]) + getElo(row[14]))/3

Ea = 1/(1 + 10**((averageBlueElo - averageRedElo)/400))
Eb = 1/(1 + 10**((averageRedElo - averageBlueElo)/400))

if int(row[7]) > int(row[8]):
Sa = 1
Sb = 0
elif int(row[8]) > int(row[7]):
Sa = 0
Sb = 1
else:
Sa = 0.5
Sb = 0.5

dRa = kFactor*(Sa-Ea)
dRb = kFactor*(Sb-Eb)

setElo(row[9], getElo(row[9]) + dRa)
setElo(row[10], getElo(row[10]) + dRa)
setElo(row[11], getElo(row[11]) + dRa)
setElo(row[12], getElo(row[12]) + dRb)
setElo(row[13], getElo(row[13]) + dRb)
setElo(row[14], getElo(row[14]) + dRb)

calculateElo('matches_week_2_2014.csv', 24)
sortedTeams = sorted(teams, key=teams.get, reverse=True)
for team in sortedTeams:
print(team, getElo(team))

[Ether]

The usual Twitter data caveats apply :)

[Joe Ross]

How do the results change if you don't include eliminations?

[connor.worley]

Teams that were upset in elims move up a bit (most notably 1114 with a 35 spot jump). Here's a quick look at the top 25 without elims:

Team Elo Net change
1678 1348.704233 0
1817 1344.119366 3
1024 1343.320635 12
4450 1327.675696 36
33 1324.168502 5
987 1322.929154 13
2468 1320.809335 -4
3393 1319.576459 12
624 1315.602349 -2
1114 1315.295585 35
3683 1315.061656 36
16 1314.684274 -6
1403 1314.394943 18
3604 1313.925403 36
2337 1313.398484 -11
118 1313.160747 -7
4488 1312.959919 -6
176 1312.405035 9
525 1310.636828 -7
254 1310.254616 -18
1736 1305.365745 34
1023 1303.956975 49
1876 1303.95267 46
294 1303.850808 22
3009 1301.648885 27

[Michael Hill]

I took a look at this last year just for giggles. Included is perl code I used. Also note that I used a k-factor of 32. I normally don't concern myself with rankings/OPR/etc., but I was bored one day.

This thread combined with watching Searching for Bobby Fischer earlier today got me wondering how to apply the Elo Rating to FRC, and I think I figured it out. This is the resulting data from the 7 weeks of the 2013 season (Sorted by rating)


4650 982.089794296798
4755 998.879722986273
1160 1005.63580696492
4234 1007.80265488653
321 1007.93578705957
4060 1014.18233417492
865 1019.24769809173
204 1020.69112670188
818 1026.19352966173
4398 1026.94694016844
1506 1027.13742298458
3167 1028.83751255313
4573 1029.51467487536
3607 1031.21268304486
3453 1031.409363359
2506 1032.43385656209
4532 1032.58573597814
4471 1033.6777037715
1255 1036.95596581586
3515 1037.28640791865
2902 1037.42220102578
4081 1037.84181433329
4680 1039.48464889698
2171 1040.8630633741
4743 1041.31651481374
2926 1042.093260529
451 1042.62182460209
322 1043.6491404653
2832 1045.7148780529
4517 1047.40106468987
4032 1048.18782072589
581 1049.54403251529
4404 1052.80909969208
3536 1053.85665934844
4653 1056.60256481931
1941 1057.03537072986
3256 1057.419034185
4289 1060.95837747809
2853 1061.25162605869
3302 1061.69907895085
3237 1062.41655580015
3783 1062.87822077035
4454 1063.26768643071
4406 1063.49864842402
1922 1064.38666730322
1522 1064.83589998138
4793 1066.85041465773
3493 1067.18136428509
1675 1067.33292602519
2405 1067.96609174255
1351 1068.64718936007
1661 1069.05052097351
3491 1069.21829793876
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3719 1069.61730092672
4337 1070.04721840224
2180 1070.50167915705
1607 1071.82120266813
280 1072.28193752885
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907 1072.77895957834
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3831 1074.83144694104
423 1074.93771105545
4387 1075.14320913319
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3259 1077.66200955262
313 1077.96750350088
752 1078.16212218756
3640 1078.17050159868
2156 1078.77450505749
4373 1079.01398736869
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3055 1080.49350385937
3100 1081.2064890464
1209 1081.44587036605
2458 1081.69975698823
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548 1082.03753483035
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4088 1083.3464915471
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857 1089.02084028869
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4753 1091.05346478885
3603 1091.61933041254
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1918 1438.35183688243
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1937 1214.33288266654
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1965 1166.03823016061
1967 1266.59804657597
1972 1109.94716524097
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1977 1246.6259697767
1982 1239.5846866044
1983 1540.37859616682
1984 1176.79889908095
1985 1404.12577305486
1986 1696.82704216305
1987 1243.19824352584
1989 1083.64787319436
1991 1259.25870102443
1994 1106.97069886758
1997 1144.12092370351
2000 1433.42748392377
2001 1170.97728561531
2002 1207.15090570157
2004 1146.78066808963
2010 1122.64224177735
2013 1330.84744303641
2016 1304.77158226126
2021 1231.36218991602
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2027 1171.82541994674
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4685 1115.30074192017
4687 1215.45359326252
4688 1194.4065205897
4689 1198.19776426133
4690 1225.62561963749
4692 1212.94881720966
4693 1199.22499109885
4694 1171.09005869826
4695 1167.09105944207
4696 1163.30490059229
4698 1165.21144465656
4701 1200.69060532397
4702 1196.00629027072
4703 1141.26576093999
4704 1201.0762811594
4705 1140.13884281152
4706 1233.44015640996
4707 1159.75504933466
4708 1186.23424144233
4711 1139.51564686851
4713 1138.26572949528
4714 1169.61538872201
4715 1200.95033231691
4716 1260.81704415184
4717 1169.65851004199
4718 1209.93014842886
4719 1127.29336875316
4720 1121.01128511953
4721 1069.35729443129
4722 1143.61218925218
4723 1125.85270404079
4724 1197.19823703816
4725 1126.44849672692
4726 1100.41776108631
4727 1186.43715536608
4728 1168.14737293399
4730 1166.06356698833
4731 1229.0200386629
4732 1162.29188833652
4733 1225.61405873162
4734 1201.08206473769
4735 1149.7482810424
4736 1214.57824569301
4737 1193.76150791181
4738 1197.1615147755
4740 1244.44169143776
4741 1201.91186017079
4742 1168.07650716011
4743 1041.31651481374
4744 1188.55844975682
4745 1145.15967413172
4746 1173.27775251891
4747 1081.86246365315
4748 1112.85552992071
4749 1221.35842325523
4750 1106.26307329814
4751 1216.12189968493
4753 1091.05346478885
4754 1187.98476245368
4755 998.879722986273
4757 1156.11347032437
4758 1209.71817951448
4761 1185.59054362967
4762 1185.47756780316
4765 1158.31665070312
4767 1127.8118361406
4768 1116.40533115538
4769 1136.78407372899
4770 1155.50193681833
4771 1137.93171352201
4772 1225.33534018324
4773 1108.35835814322
4774 1102.86871945862
4775 1181.64706221452
4776 1207.35075431759
4777 1145.30582631384
4778 1144.02463726571
4779 1092.33068291562
4780 1165.47494331293
4781 1149.71598385133
4782 1072.53570071915
4783 1177.94184249394
4784 1142.50149603286
4785 1077.39028590692
4786 1119.61577576642
4787 1108.26229647447
4789 1180.85127323254
4790 1183.42877599409
4791 1226.77289361863
4792 1214.42095830569
4793 1066.85041465773
4794 1181.75000684866
4795 1146.03828902451
4796 1113.61698112973
4797 1332.99162879393
4798 1165.40813671457
4799 1197.62673412403
4800 1211.17511453921
4803 1214.76163042569
4804 1124.79248490191
4805 1255.73839796082
4807 1102.18542834292
4808 1138.31499585144
4809 1202.24581825252
4810 1119.52092376664
4811 1121.03119282398
4812 1252.25141636152
4814 1214.25999977625
4815 1189.85870239641
4816 1184.70089881474
4817 1199.68445669747
4818 1152.53475264832
4819 1230.46897335443
4820 1182.27527319276
4821 1210.0816360906
4823 1182.48251964251
4824 1274.00412127724
4825 1147.01795295115
4827 1148.50518719898
4828 1278.93111983802
4829 1210.37691627317
4830 1133.17375468716
4832 1191.36321864805
4833 1123.88222989458
4834 1095.55371195256
4835 1123.55145563676
4836 1138.62509667444
4838 1132.83470089359
4839 1146.96341090907
4840 1166.11151450111
4841 1164.16204016054
4842 1221.94635528239
4843 1132.06854216563
4845 1198.5312620239
4846 1123.13300295534
4847 1185.98943767919
4848 1204.42495894197
4849 1096.67558771439
4850 1214.51286765616
4851 1132.70232544743
4852 1173.6052343879
4853 1169.5893564853
4854 1126.19486187291
4855 1241.04231318184
4856 1101.63239304157
4859 1140.91825536466


Here's the Perl script I used. It takes a standard input file of a match number (just sequential is fine), red teams, blue teams, ("RED"|"BLUE"|"TIE") (referring to who won.

#!/usr/bin/perl

my %rating = ();

my @data = <STDIN>;
my $K = 32;
my $defaultElo = 1200;


# Just make sure a team number and default value of 1200 exists for each hash.
foreach(@data) {
my ($match, $red1, $red2, $red3, $blue1, $blue2, $blue3, $res) = split;

if (! exists $rating{$red1}) {
$rating{$red1} = $defaultElo
}
if (! exists $rating{$red2}) {
$rating{$red2} = $defaultElo
}
if (! exists $rating{$red3}) {
$rating{$red3} = $defaultElo
}
if (! exists $rating{$blue1}) {
$rating{$blue1} = $defaultElo
}
if (! exists $rating{$blue2}) {
$rating{$blue2} = $defaultElo
}
if (! exists $rating{$blue3}) {
$rating{$blue3} = $defaultElo
}
}

foreach (@data) {
my ($match, $red1, $red2, $red3, $blue1, $blue2, $blue3, $res) = split;

my $Ered = 1 / (1 + 10**((($rating{$blue1}+$rating{$blue2}+$rating{$bl ue3})/3-($rating{$red1}+$rating{$red2}+$rating{$red3})/3)/400));
my $Eblue = 1 / (1 + 10**((($rating{$red1}+$rating{$red2}+$rating{$red3 })/3-($rating{$blue1}+$rating{$blue2}+$rating{$blue3})/3)/400));

if ($res eq 'RED') {
$rating{$red1} = $rating{$red1} + $K*(1 - $Ered);
$rating{$red2} = $rating{$red2} + $K*(1 - $Ered);
$rating{$red3} = $rating{$red3} + $K*(1 - $Ered);

$rating{$blue1} = $rating{$blue1} + $K*(0 - $Eblue);
$rating{$blue2} = $rating{$blue2} + $K*(0 - $Eblue);
$rating{$blue3} = $rating{$blue3} + $K*(0 - $Eblue);
}
if ($res eq 'BLUE') {
$rating{$red1} = $rating{$red1} + $K*(0 - $Ered);
$rating{$red2} = $rating{$red2} + $K*(0 - $Ered);
$rating{$red3} = $rating{$red3} + $K*(0 - $Ered);

$rating{$blue1} = $rating{$blue1} + $K*(1 - $Eblue);
$rating{$blue2} = $rating{$blue2} + $K*(1 - $Eblue);
$rating{$blue3} = $rating{$blue3} + $K*(1 - $Eblue);
}
if ($res eq 'TIE') {
$rating{$red1} = $rating{$red1} + $K*(0.5 - $Ered);
$rating{$red2} = $rating{$red2} + $K*(0.5 - $Ered);
$rating{$red3} = $rating{$red3} + $K*(0.5 - $Ered);

$rating{$blue1} = $rating{$blue1} + $K*(0.5 - $Eblue);
$rating{$blue2} = $rating{$blue2} + $K*(0.5 - $Eblue);
$rating{$blue3} = $rating{$blue3} + $K*(0.5 - $Eblue);
}
}

while(($key, $value) = each(%rating)) {
print $key . "\t" . $value . "\n";
}


Not sure how useful the Elo ratings are, but there is some statistical significance with having teams like 469, 67, 1114 and 2056 near the top. I'm nowhere near good enough with statistics to determine if this is enough data to work with (my instinct says no), but it's just another interesting way to look at wins/losses

(Edit: Had mistake in my calculation)

[Ether]

Included is perl code I used.

Here's Connor's translated to GNU AWK 4.0.1. Input is standard export.csv file.



BEGIN {FS=","; ln10=log(10); kFactor=24;}

(NR>1) && (($4=="Q")||($4=="E")) {

for(i=8;i<=13;++i){if(!($i in Elo)){Elo[$i]=1200}}
avgRedElo = (Elo[$8] + Elo[$9] + Elo[$10])/3;
avgBlueElo = (Elo[$11] + Elo[$12] + Elo[$13])/3;

x = ((avgBlueElo - avgRedElo)/400)*ln10; Ea = 1/(1 + exp(x));
x = ((avgRedElo - avgBlueElo)/400)*ln10; Eb = 1/(1 + exp(x));

if($6>$7){Sa=1; Sb=0} else if($7>$6){Sa=0; Sb=1} else {Sa=0.5; Sb=0.5}

dRa = kFactor*(Sa-Ea);
dRb = kFactor*(Sb-Eb);

Elo[$8]+=dRa;
Elo[$9]+=dRa;
Elo[$10]+=dRa;

Elo[$11]+=dRb;
Elo[$12]+=dRb;
Elo[$13]+=dRb;
}

END{
PROCINFO["sorted_in"] = "@val_num_desc";
for (TeamNum in Elo) {print TeamNum "," Elo[TeamNum]}}

[Michael Hill]

Free good rep to the first person to post FORTRAN or COBOL code of this.

[Andrew Schreiber]

Free good rep to the first person to post FORTRAN or COBOL code of this.

Bonus points for shortest viable MUMPS (http://en.wikipedia.org/wiki/MUMPS) implementation :)

[Ether]

Free good rep to the first person to post FORTRAN or COBOL code of this.

How about Pascal? Would you give reps for that?

[Michael Hill]

How about Pascal? Would you give reps for that?




lol of course not :p

[Redo91]

I was curious how the data looked. I just threw it onto Google Docs(after making it into an Excel file because Docs does not like that amount of characters being directly pasted into it). Here it is. (https://docs.google.com/spreadsheet/ccc?key=0As1gaKYCmf5ldFR6YU41RWFKTUhtX05MWHpBeVYtS 3c&usp=sharing)

Edit: Realized this isn't terribly relevant after rereading the thread title. Still, a good visual I think.

[Redo91]

Here (https://docs.google.com/spreadsheet/ccc?key=0As1gaKYCmf5ldGZOaC1sR2NHNWszSGswQkp6RG55U kE&usp=sharing) is a visual using the data for the first 2 weeks using the data from the paper.

[Michael Hill]

I was somewhat bored this evening and decided to try something different. I decided to try the Microsoft TrueSkill algorithm to rank teams. If you're unfamiliar with TrueSkill, it's the algorithm used on XBoxLive to determine fair matchups. It's based on the Glicko rating system, which was developed as an enhanced Elo rating. Glicko introduces Bayesian statistics to the Elo system and TrueSkill takes it a step further and includes team-based play. The results were...interesting. The first column is team #, second column is the team's skill rating, the third is sigma (standard deviation)


3478 32.1901547 3.20363857
1678 32.0112708 2.399343622
1319 31.87906344 3.315070058
1619 31.75256228 3.470732969
2912 31.58129439 4.715476202
4509 31.40973953 5.133444261
2363 31.31808794 2.836302053
118 31.29619384 2.538432629
1230 31.15380747 3.547815035
2056 31.14569129 2.576563062
1718 31.11263159 2.109044982
195 31.10461402 2.192780006
254 31.04528151 2.330684176
967 31.01357444 3.745206881
225 30.96307625 1.973250808
4539 30.94319405 4.865262985
1310 30.83621852 2.755701999
870 30.83507163 3.272737755
16 30.81035963 2.664498783
1114 30.80861686 2.360223
695 30.77693275 4.33806769
4415 30.77316599 4.730511852
79 30.75791237 2.826369003
1987 30.67232661 3.625521206
1982 30.60625983 4.670991754
383 30.59868127 4.8017054
2485 30.59596864 2.743650604
1410 30.45837899 3.416039844
148 30.43365623 2.481575794
3741 30.41991712 4.512545091
3102 30.41881484 5.213708006
3472 30.40490716 4.859007573
3612 30.3931785 3.722623394
3658 30.37491946 4.472326569
294 30.33797426 2.756791626
971 30.30915264 2.618289057
180 30.30718968 2.695798753
1736 30.20581023 3.308573931
354 30.2000696 5.039422075
1023 30.19938536 2.035922773
3834 30.14868708 4.586989693
2614 30.12043279 2.804402804
696 30.10697699 3.694183464
334 30.08551724 2.806576511
4930 30.0675904 2.735389892
1519 30.05333958 2.241897536
1323 30.00312004 2.643458512
4587 29.99732998 3.348657276
3824 29.98859858 2.932023783
5089 29.95855914 4.922216515
2809 29.92320705 3.297946178
3393 29.8910423 2.20798222
5184 29.88130369 4.091101226
4913 29.8774892 4.678576218
3193 29.86982285 3.311722746
1717 29.85482878 2.7075201
67 29.8301642 2.09103307
4096 29.80669386 4.516803273
2883 29.79046227 3.722170155
3009 29.78901767 3.375059235
3794 29.76813191 3.217671954
33 29.67701471 1.932827532
1318 29.64508078 2.003948916
3038 29.63862012 4.982429233
1775 29.63055641 2.76632969
624 29.61103972 2.557791883
399 29.61042143 2.40927544
4188 29.60319508 5.072906277
231 29.5981198 3.075990159
3981 29.57025821 4.737150581
2388 29.53343602 3.505482723
3604 29.50671464 2.387871186
3507 29.41604642 3.473038332
4238 29.40460167 5.128627995
2052 29.39817072 2.852137471
3098 29.37634231 2.100343943
1730 29.35808437 2.866267364
4705 29.35425678 4.328548166
2481 29.34448721 2.597287993
5160 29.3410846 4.407520664
131 29.33269732 2.410860812
461 29.2949643 3.400288017
27 29.2584181 1.963683284
836 29.23384075 2.812459901
1986 29.21831157 2.788409232
2090 29.20563236 4.667521749
5057 29.20529178 3.066809556
610 29.20137097 2.730510579
1825 29.17707902 4.513578798
4028 29.16935309 3.236107507
3227 29.16366806 3.576224541
3386 29.13015577 2.805551126
5261 29.12090884 4.52024528
3018 29.11684387 2.780230959
314 29.1137908 2.155851246
2607 29.10596373 2.474612815
3491 29.09457281 4.732290359
2451 29.08057067 2.705712664
846 29.05322417 2.708043561
401 29.03981884 3.68830887
138 29.03914468 2.433997114
2137 29.0239724 2.096970259
4967 29.0201646 2.121102898
1583 29.01446561 5.085032027
4250 29.00906134 4.589249874
1396 29.00411684 4.983703554
2996 28.99853297 2.637380121
4003 28.99627635 2.418860007
145 28.9865935 4.026913446
1658 28.98607397 4.378878932
2797 28.9772021 5.011390959
364 28.97658572 3.260667401
4911 28.97471817 1.959958519
5289 28.96901789 5.116254183
1164 28.96780246 4.41807995
1165 28.96298327 3.667415147
2613 28.95773029 4.522513711
51 28.95283117 2.04568091
330 28.94137205 2.757355386
2550 28.928286 3.259926747
1065 28.92414986 3.350546227
1418 28.91922536 3.625238161
1245 28.91299834 4.696817574
3299 28.89864893 4.77036342
2122 28.88314202 2.632342013
2590 28.87763451 2.12109764
4050 28.87537735 4.4408568
2337 28.86109549 1.947189895
1075 28.8589027 3.486470276
2059 28.85801696 3.855572027
3056 28.84388821 4.698165799
1640 28.84385094 2.115164223
2472 28.83584063 3.937474327
4265 28.82053179 3.202783641
2470 28.81843083 3.62434634
2200 28.80200222 3.454150345
2853 28.80057617 4.532767299
3683 28.78703528 2.573634595
5172 28.78531008 3.17008341
1592 28.76603843 2.780417592
4039 28.76053783 2.698909212
4488 28.75751498 2.066478142
1218 28.75058556 1.965513343
3230 28.74965528 2.98843785
3158 28.74459617 2.919053474
5290 28.74032139 4.986355668
842 28.7384048 3.390498681
2875 28.73486068 4.393809797
1937 28.72667953 3.159170361
3939 28.72377262 5.014530397
1477 28.72195481 2.645931196
1625 28.71449008 2.572107155
359 28.7106173 2.428574051
558 28.70245237 1.966845141
2508 28.7017193 4.867675197
639 28.69219888 2.793132014
494 28.68548348 2.008560481
346 28.68508561 3.701458387
2528 28.67792171 3.650580436
744 28.66809808 3.423887217
1421 28.6531687 3.758194739
4551 28.64865284 4.168401925
75 28.63578832 2.121493199
3467 28.62975734 2.139867106
2907 28.62606271 2.222687386
2761 28.61954732 3.39224997
2068 28.61136406 3.887963826
3211 28.60418949 3.146403127
3334 28.59815921 4.34943834
5349 28.59570059 4.466291466
1742 28.58203823 4.728482913
3259 28.57764535 4.784459804
3189 28.55687292 3.467863461
2928 28.55016722 1.896961764
4612 28.54101822 4.666243576
339 28.53098874 3.602334025
1817 28.52943668 2.62451883
1262 28.51584333 4.714753286
60 28.51465696 3.658851444
4522 28.51327424 2.496180895
192 28.51009496 2.837694214
1507 28.50904353 2.720309797
250 28.50737526 3.218727996
3525 28.50580939 1.929654217
1756 28.50316361 2.730433292
3044 28.4997126 3.177095472
222 28.49367617 2.025079777
2339 28.48436254 3.301338552
540 28.47933421 3.217824853
862 28.46126207 2.023734676
3389 28.46099026 4.68596841
1379 28.45657369 5.195424462
1675 28.439144 3.261421717
2156 28.43537317 4.275714783
4462 28.42975519 4.550973536
70 28.41177903 1.895989771
135 28.408739 2.636827712
155 28.40706035 2.688336749
469 28.40096089 1.874391042
2183 28.39996952 4.687026597
2078 28.3977569 4.833904451
319 28.39395756 2.083071777
58 28.36280402 2.087280415
3206 28.35801911 3.688675189
3998 28.34404697 4.849336928
1257 28.32025459 2.499704853
3418 28.30933828 4.349595634
3388 28.30259473 4.824801315
1285 28.29458059 2.305152996
597 28.27834188 2.787716954
447 28.26467088 2.561633227
486 28.25232446 2.973760953
3015 28.24856616 2.602491856
3735 28.23414165 3.150216293
1311 28.23280496 2.804190221
1731 28.23135219 2.845290234
246 28.21585028 2.431129206
2107 28.21525663 4.703143024
4266 28.21192071 4.778905794
772 28.20214026 2.383205586
86 28.19176133 3.376964429
3005 28.19093489 4.501310307
4400 28.18151359 4.64052433
2486 28.17937362 2.805844071
3117 28.16664785 3.425938594
3656 28.16562902 3.323462653
3539 28.15919773 1.991295532
1329 28.15724567 3.355881553
2013 28.15667815 2.378503565
1876 28.1516167 3.762993367
5005 28.14456079 4.575925432
3535 28.14181724 2.936168852
3822 28.14036599 4.673917786
1828 28.13139891 3.701192659
4451 28.12510102 3.823729966
2181 28.12212267 4.875266253
2641 28.11901782 3.091051845
4092 28.11055919 3.751631445
125 28.09033155 1.853441856
4498 28.0901825 4.465208775
3572 28.07534664 2.859355582
3588 28.0750612 2.557894259
5282 28.07459544 4.17573687
3482 28.07399642 3.389302297
1111 28.07010785 3.632608132
3268 28.06727843 2.83892901
188 28.06298756 2.656421278
4070 28.05912621 4.502580614
107 28.05697432 2.145286865
4946 28.05596303 2.646533546
4158 28.05315935 4.618358386
3090 28.05005009 4.717190071
3003 28.0498165 3.274335467
2935 28.04793134 4.435358283
1816 28.02961966 2.862994721
4732 28.02384662 4.036594982
1942 28.01633271 4.863433025
1280 28.00219279 3.260433478
279 27.99347211 3.230759844
3763 27.99139831 4.447420555
4073 27.97968297 5.065743872
1885 27.97598351 2.7666224
4085 27.97469156 3.471272455
3614 27.97216098 4.960212238
2190 27.95430507 3.930074667
900 27.94524399 2.707364614
4820 27.9390299 4.688622033
21 27.93410091 3.432868247
3526 27.93169239 4.814848781
1538 27.92486838 2.724456636
857 27.92354123 2.098624606
5023 27.92203199 4.171843831
4656 27.92141755 4.80487981
4598 27.91955858 4.481995239
304 27.90909841 2.322057528
2375 27.90894942 3.603518954
2202 27.90734135 2.633259675
2959 27.89732735 2.007381819
4153 27.89122899 3.463395154
2252 27.88971547 2.756102993
228 27.87601748 2.109778856
3504 27.87224357 2.759226657
1317 27.8666563 4.832387572
78 27.86462222 2.117100866
4828 27.8595393 4.812909586
3620 27.85597677 2.080220628
1025 27.8536183 3.129743052
93 27.85089194 3.633570165
3944 27.84765204 5.019998598
2338 27.84209635 3.279898247
876 27.84083004 3.451462062
1596 27.83996243 3.067789924
3310 27.83584393 2.745422584
1747 27.83319892 3.462503992
4111 27.83258837 4.96614408
2969 27.82606189 5.049974036
343 27.82084209 4.049864518
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4484 26.0548508 4.455726089
2004 26.05453772 4.763668828
4633 26.05020964 4.005286843
1815 26.04983267 4.663266242
3534 26.04706623 2.388449645
245 26.0453876 2.115680298
1750 26.04293435 4.76870836
973 26.03913264 2.714604862
4283 26.03424007 3.129550525
4601 26.03215301 3.537598454
2423 26.02399272 3.052377281
3765 26.02091945 5.133492585
4127 26.01507353 2.449883627
5302 26.01348526 4.193129223
4910 26.01214001 4.956101754
1952 26.00823002 4.829085143
555 26.00738109 2.475346292
4513 26.00516059 3.113976573
2439 26.00449134 3.397106684
3958 25.99785175 3.419857228
1438 25.99394787 4.655325771
3340 25.99028638 2.859971601
2177 25.98959104 2.858639482
4139 25.98772837 3.348619787
1157 25.98318857 4.823218855
1014 25.97303174 3.238123515
4653 25.96958585 2.488406342
3103 25.96586623 2.722495932
4952 25.96320817 4.8488014
4381 25.95668772 2.672570854
2574 25.95100705 3.689420592
743 25.94629998 3.725348569
2158 25.9406868 2.803883339
829 25.93858261 3.335568116
4542 25.93218408 2.268942689
470 25.92900546 2.712770523
848 25.92858127 4.667055468
5299 25.92656074 3.336281888
3017 25.9264484 3.698140148
4464 25.92180028 2.768175565
61 25.92141899 2.296819806
2415 25.92068257 2.670833009
1569 25.90952041 4.208240308
3138 25.90810927 2.651098824
3861 25.90127159 4.812618119
4914 25.89795092 2.765773908
3294 25.89646384 5.0367673
280 25.89635151 2.416139724
3241 25.89501023 4.294856632
930 25.89292182 4.26501343
4568 25.88869622 3.274545025
3989 25.88591812 4.661304976
237 25.88168813 2.540944845
3542 25.88074749 3.235241647
3288 25.87907725 3.338059997
111 25.87734569 2.620246822
5136 25.87421649 3.233836853
1741 25.87037265 3.519553984
340 25.86990997 2.582688486
2848 25.86403163 2.690655936
4456 25.85881388 3.561056191
4247 25.85659721 4.230306483
3292 25.85590202 4.918602463
4352 25.8499675 4.971420091
3451 25.84979221 3.094023398
4048 25.84728806 2.267019922
204 25.84612418 2.517053417
1700 25.84127525 3.175144409
811 25.83803075 2.12468327
4175 25.83384099 4.349167177
4726 25.83320278 2.977810115
4676 25.82594744 4.315738267
316 25.82481492 2.273709941
5161 25.82327837 3.045787057
1891 25.82166143 4.099315737
1739 25.82139118 3.419477996
2910 25.81443519 3.374864624
4201 25.81330262 3.401463618
4343 25.81215503 3.37375928
3352 25.80437606 3.441060808
999 25.80430157 2.350787378
48 25.80258738 2.382335397
1692 25.80172521 4.624153001
4849 25.79694313 4.610311768
2704 25.7942631 3.319372868
2457 25.79075761 4.666099819
2149 25.79020623 2.41177977
453 25.78789702 2.38818413
3881 25.78662744 3.441167939
810 25.78404993 3.730171249
1350 25.78399234 2.841195365
4800 25.78117337 4.580236285
2869 25.78019509 4.878641097
3709 25.78002205 4.793501878
34 25.77725812 4.751276857
3555 25.77571947 3.483371002
4027 25.77183415 4.486230455
4091 25.76451189 4.697675647
4624 25.76377657 4.695358626
835 25.76232095 2.970896703
606 25.7620557 4.570989037
2823 25.76194156 4.880508492
1796 25.75828223 3.717976596
3965 25.75750696 3.561090342
4694 25.75480253 4.416583686
4534 25.75381381 4.699388735
2973 25.7486724 3.748649123
3792 25.74617129 4.235462911
3481 25.7456566 3.04631725
1250 25.74391386 2.37587626
4777 25.74319086 4.04647475
2531 25.74141852 4.632646496
1761 25.73871853 2.953334359
5230 25.73860625 2.963977863
100 25.73487455 3.133382358
1137 25.7336749 4.887332239
5003 25.73248562 5.020100817
4717 25.72964521 4.582595251
1261 25.72858201 2.81889641
2584 25.72410757 4.638031775
1633 25.72390211 4.99450338
4384 25.72351402 3.085015565
4973 25.72267697 4.647138626
1073 25.71769281 2.365944176
3357 25.71614237 3.66067293
2526 25.71370525 5.007304738
378 25.71205725 2.638254401
4102 25.7117055 4.437586924
1361 25.70598967 4.821173608
2084 25.70585107 2.541179523
5144 25.69921737 3.005643551
2992 25.69783516 3.653019213
3561 25.69536358 4.983700967
3460 25.6931703 4.523746471
3374 25.69132795 4.301707291
178 25.69052589 2.1545598
3571 25.68758326 3.648516709
5012 25.68651749 2.731842173
1989 25.68614161 2.460306339
501 25.68455023 2.979488487
3140 25.68404945 3.717867335
386 25.68116149 3.42413329
1502 25.68024141 2.676003794
2847 25.67984833 4.755306211
1553 25.67958708 4.87999649
2603 25.67765839 4.79711919
3999 25.6753191 3.705666205
4928 25.67526694 4.753734617
4795 25.67439086 4.610460528
4255 25.67175231 4.260518063
2890 25.67020932 4.649282044
1758 25.67001989 3.664607988
538 25.66820431 4.57451238
3255 25.66538562 3.443509708
4473 25.66388121 2.206872697
358 25.66330158 3.064885593
3660 25.65773871 3.067528095
4646 25.65409878 3.544512616
4461 25.65399259 3.272188888
4264 25.65391465 4.645461031
4135 25.64285596 3.512537247
4827 25.64059679 3.075858951
2198 25.64011025 3.297096596
1711 25.63735726 2.345621307
5340 25.63597618 5.000337098
3767 25.63561587 2.987549523
4765 25.63508275 4.210027308
4011 25.6342395 4.853041648
4625 25.63166803 4.133051859
4009 25.63099902 4.956826279
4593 25.63064101 5.017402572
5190 25.62712305 3.201753278
2187 25.62662565 4.7130277
4782 25.62645409 4.784694553
3868 25.62476913 4.402621424
4549 25.62252939 3.692639183
2972 25.62229042 3.366534934
3882 25.62220325 4.774745384
4954 25.62022793 2.372455853
1927 25.6194747 5.130251585
1984 25.61886177 4.715334182
3760 25.61092496 4.751900681
2395 25.60922567 4.453403738
858 25.60623625 3.125635574
5119 25.60162795 4.795939761
1302 25.60032367 2.264116783
3320 25.59706068 5.097648999
2192 25.59699299 3.004169133
5143 25.59644426 5.079989327
2929 25.59557055 3.2731385
5242 25.58995457 4.51154077
5106 25.5875175 4.677039755
1033 25.58050913 4.874723116
433 25.57937821 2.372069504
5145 25.5793524 3.279545023
1197 25.57291957 3.522335782
5166 25.57100763 3.256311916
4110 25.5700043 2.78098605
3617 25.5697536 3.22227272
4575 25.56901276 2.821268881
5251 25.5666995 4.422932374
3142 25.56619221 2.267311135
1156 25.56496852 2.82018508
3512 25.55801056 3.484010787
3546 25.55549339 3.776975241
839 25.55411759 2.875154711
3382 25.55147892 4.618733376
4054 25.55129373 5.124226428
3128 25.54854102 4.500585932
509 25.54527492 3.12314446
2909 25.54156456 4.668075495
1011 25.54069887 2.92962792
3461 25.54049723 2.408429052
4586 25.54009874 5.110525881
2632 25.53431784 4.734946682
2340 25.53370406 4.451863918
1695 25.53108432 4.254132937
3167 25.53047881 3.118857406
1737 25.52127245 4.508238166
1701 25.51975284 2.623468044
1195 25.51687843 4.811670092
2525 25.51541663 4.689395977
4004 25.51445947 3.018323443
5031 25.51333323 4.725898252
4490 25.51211312 4.642831294
171 25.50981405 3.360607851
4457 25.50693538 2.428362557
3880 25.50693169 3.238465311
2383 25.50315156 3.016170818
1899 25.49914579 3.030103074
2635 25.49893633 2.393166188
3324 25.49890784 4.070878561
2499 25.49797659 5.12070412
5276 25.49703342 4.800001084
2239 25.49331524 4.828035019
3684 25.48940875 3.292290666
4680 25.48939289 2.844312264
2980 25.48424865 2.176341854
2976 25.48328256 2.53049296
1719 25.47842295 3.647722572
293 25.4721076 2.352276952
108 25.47022506 3.558229018
5137 25.46616955 3.103568926
1429 25.46267481 3.493153169
3714 25.46138919 4.627633861
342 25.45573477 3.658140807
2374 25.43979125 2.97333266
2551 25.43860263 4.234717662
2974 25.43642674 2.896112102
2493 25.43566144 3.565427759
5191 25.42986489 3.025447621
2537 25.42866873 3.669773188
670 25.42829303 3.520492582
3946 25.42632596 4.86296507
5162 25.42608084 2.628944356
4290 25.42491821 4.51777209
3711 25.42310084 2.358362282
2498 25.42248088 4.961637635
3961 25.4224772 3.708770861
5111 25.42160592 2.364452459
5194 25.41988109 3.28110173
587 25.41771344 3.599839454
1577 25.41729918 4.963829127
3164 25.41628042 3.381147794
1983 25.41589346 2.113843094
95 25.41345456 3.228008016
884 25.41259937 3.026332984
704 25.40731064 3.513257631
2403 25.40076238 2.94738915
57 25.40040438 3.631456184
3387 25.40012204 3.700419082
2001 25.39735382 4.531829645
3826 25.39481891 3.418292776
371 25.39277366 3.547696623
1908 25.39163255 4.773467765
2611 25.39101768 3.177697682
63 25.3858905 3.660390074
2871 25.38325119 3.062227278
2990 25.37886633 2.410159052
1100 25.378469 2.288962532
3234 25.37804004 2.983542187
3793 25.37609307 4.63839743
4908 25.37565098 2.421141633
4014 25.37494531 4.670375908
4226 25.37454845 4.812482248
1551 25.37441359 4.244952042
686 25.36663015 3.690416809
2170 25.36634262 3.338661655
2512 25.3645518 3.397113917
4327 25.36294792 2.976102184
4012 25.36237554 5.029580492
4573 25.35857736 2.527534177
2444 25.35608656 3.23631681
2897 25.35454212 4.728974911
2199 25.35088002 3.47961028
2473 25.33994942 3.261003996
4332 25.33660887 4.96335051
5248 25.33606417 3.158157544
3238 25.33555991 2.097925259
3100 25.32833197 4.700940429
2673 25.3273103 3.001290433
291 25.32580941 3.310465304
5039 25.32145135 4.041817807
3548 25.32047822 2.527573441
4582 25.32046177 4.932474124
3659 25.31966047 4.868668136
122 25.31946069 3.638404347
4559 25.3189624 2.307153825
2080 25.31894942 2.816082278
3039 25.31483966 5.081277443
2936 25.3118763 3.669333472
3236 25.30941479 2.723351559
3602 25.30738924 2.145883001
709 25.30690616 3.156428673
3219 25.30567049 2.399956041
5053 25.30254219 2.29325552
5164 25.30131476 4.688192245
1846 25.29832584 3.377036114
2504 25.29488016 4.670289545
1160 25.29380764 3.425885914
3568 25.29261973 3.181554282
4567 25.29029893 4.541366246
5082 25.2902268 4.398354218
4023 25.28349664 4.060374707
3623 25.28284922 2.937444403
4232 25.28152901 4.208384721
2225 25.27972522 4.949354816
5048 25.27817212 2.826419106
3075 25.27407971 4.967746554
4171 25.26749566 2.991079168
4499 25.26749373 2.638582727
4376 25.26729379 3.059815272
1378 25.26686127 4.460640773
2729 25.26273859 2.495680489
4361 25.26244816 2.420830311
2411 25.2623798 2.308969889
5247 25.26080602 3.026012299
2185 25.25980363 4.46862356
353 25.25959175 2.78269167
3373 25.25931466 3.792342893
4382 25.25473672 3.210739303
4418 25.25135613 4.640494372
333 25.25114012 3.725479599
4069 25.24575324 2.657666584
1018 25.244359 3.255882957
3543 25.23747806 4.41785509
980 25.23155638 3.428655255
1751 25.23018315 4.446594733
1126 25.2301364 2.5089232
3204 25.22546874 4.97841685
4855 25.22512113 2.877373895
1038 25.22446844 3.178167514
3132 25.22032605 4.668640599
2549 25.21494944 4.731472717
4761 25.21251303 2.414999155
4501 25.21223891 4.797123487
103 25.20760538 2.710565465
5350 25.20539132 4.333647605
3669 25.20538476 3.215876794
3970 25.20414766 3.627800452
907 25.20203666 3.61664582
5337 25.20202196 5.128678294
3694 25.20133029 4.960540271
2713 25.20131476 3.057428729
568 25.19664388 4.465725168
1334 25.19551663 2.248974279
1225 25.19264188 3.355911045
5060 25.18989267 4.310328874
886 25.18884998 3.616793222
1973 25.18827727 3.129999405
3293 25.18688881 4.811952097
4947 25.18632377 4.688961028
5071 25.18529285 4.40186611
3172 25.17933559 4.62954264
1517 25.17703634 2.376820059
1802 25.17608205 4.666699871
3692 25.17570888 2.863014473
3785 25.16952585 5.002419729
4956 25.16662416 2.384322295
3336 25.16660017 4.492981566
271 25.16319025 3.555468804
4556 25.16151297 4.318558996
1148 25.16058706 3.615886422
4030 25.15642213 2.505343479
3487 25.15468817 4.495361168
1554 25.154376 4.416094199
296 25.15435409 3.279221305
2539 25.15294992 2.341062686
4632 25.15095485 4.811350302
5318 25.14882964 4.852799411
4604 25.14634451 4.094997253
1720 25.14198593 4.512110861
2484 25.14052161 4.245411324
5030 25.13972407 2.868671195
2429 25.13870567 3.441139078
4980 25.13798105 2.435154286
4779 25.13713319 2.406454595
4741 25.13682861 4.902253464
5179 25.13327246 3.17570646
1388 25.13218711 3.43402703
904 25.13150944 2.563426949
2412 25.13138842 2.377255003
3240 25.13073793 5.096147469
4031 25.13072758 4.36184208
4306 25.13059602 4.540920119
4083 25.13006336 5.064400225
3987 25.12979317 4.678831663
5115 25.12893641 4.685095078
3402 25.12739263 4.393712517
3019 25.12531303 3.565186401
919 25.12123321 4.68934242
3220 25.12106708 2.98013888
4735 25.11908159 3.583106763
5066 25.11559945 3.092524373
2960 25.11401276 2.343030711
4737 25.10890684 2.907482983
2399 25.10887666 3.413913069
4958 25.10713212 2.310375513
5056 25.10645053 4.071074373
3244 25.10234335 5.084083618
4035 25.09619831 2.307710873
1208 25.09155716 3.294906782
1801 25.09109277 3.321157904
2577 25.09099786 2.418712466
3006 25.08622353 4.302683791
4353 25.08561404 4.781901243
224 25.08462731 3.090629149
4143 25.08010373 3.399611526
1708 25.07369832 3.196929101
85 25.07311621 3.129663187
1482 25.07289558 3.329739522
4906 25.07009278 2.941301455
4957 25.06949535 4.767345844
4696 25.06818925 5.210150106
4183 25.06553692 3.478494651
292 25.06471765 3.47648627
4583 25.06111871 3.178239493
1288 25.0609267 3.440170342
1369 25.05628148 3.390386591
3754 25.05573478 4.544512423
335 25.0547191 4.988289365
3559 25.05318723 4.398362279
3996 25.05230927 4.550521357
1249 25.05061257 4.5384978
3777 25.04927297 4.774348605
3971 25.04918632 4.67452444
612 25.04893417 2.841204594
2861 25.04465461 5.012595447
102 25.03758286 2.280846026
5002 25.03728475 2.646009953
5093 25.03355754 2.704847405
4541 25.02786083 3.647734377
869 25.02766234 2.121771601
2874 25.0252236 4.672053091
1991 25.02462395 2.482068074
2554 25.02337734 3.094259726
3931 25.02083681 3.380067405
4597 25.01413231 5.121432361
4521 25.0111 3.142808467
3470 25.00798659 4.388831632
1339 25.00582165 3.32895109
1635 25.00448171 4.970753579
5201 25.00401204 3.052287542
4535 24.99827103 4.48854346
3458 24.9973129 3.017142044
5193 24.99613824 2.35791344
3853 24.99598212 5.022440425
956 24.9958687 2.215603208
5151 24.99494459 4.988368507
5185 24.99220705 4.031247239
1444 24.98886838 4.444968654
5150 24.98585759 3.40996498
4390 24.97880743 2.981973393
4616 24.97780461 4.42282269
4936 24.9777217 4.03407843
3787 24.97594345 2.984670363
4013 24.97532079 3.264286682
1769 24.97334658 4.738817452
5263 24.97234634 3.205351363
4805 24.96997993 4.899428471
4277 24.96938809 4.766965538
3410 24.96850416 3.310925634
5013 24.96704701 4.634155897
3566 24.96105625 3.357861131
1325 24.96099063 2.79508404
1622 24.95607772 2.862320993
815 24.95536643 2.402270319
3403 24.9553332 4.49516814
2172 24.95362146 3.578293045
5209 24.94970467 4.677768206
4082 24.94593403 3.108100088
2915 24.94480166 3.188544527
3070 24.93652533 2.261372977
2169 24.93371678 2.939242551
1977 24.93353537 4.761371852
4905 24.93247813 2.429263588
203 24.92304602 2.418964438
4594 24.91910275 4.803150124
1631 24.914352 4.330236545
4272 24.91340652 3.36001978
4943 24.91239934 4.041702624
5077 24.90955746 4.77927719
4997 24.90757019 3.117548133
1495 24.90109263 2.31058357
3137 24.89949234 3.538033305
375 24.89809691 3.673431849
3573 24.89768752 4.772170495
4181 24.89766479 5.172683519
3013 24.89030086 4.297537147
1684 24.88794322 2.984864212
5129 24.88310098 2.847600703
1537 24.87969525 4.325331972
3008 24.87792446 2.862856716
5188 24.87748886 3.096070235
3755 24.87676411 5.00464145
5133 24.8740483 4.696656253
4918 24.87086873 2.351753821
4758 24.86900266 2.870233222
4205 24.86740215 2.744231161
968 24.86722439 4.729212611
4472 24.85843949 4.890774485
702 24.8535846 3.620605353
5307 24.84766967 4.656797865
201 24.84450239 2.605940412
3067 24.84365423 3.467586628
4404 24.84181199 4.523679248
3499 24.84036361 3.175957752
3116 24.83494458 4.568778856
4348 24.83187526 4.274751498
2370 24.82721085 2.956515879
3223 24.82544369 3.329177356
4026 24.82276545 3.400793334
4483 24.82170915 3.068729642
5199 24.81894338 4.457804819
3027 24.81763969 4.576148606
421 24.81478708 4.899150434
2461 24.81381902 3.441331181
484 24.80356558 2.016718002
2556 24.80248828 4.805890867
4662 24.80209814 2.935243661
4907 24.80091103 4.169072018
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1880 21.74772702 5.246528601
3417 21.74659217 4.508207
3157 21.7387222 4.244118035
3575 21.73826148 3.180512605
4371 21.7349426 4.558890445
5141 21.72862721 4.617924857
3361 21.72361219 5.059123885
4979 21.71848734 3.024835253
4025 21.69940462 4.592180003
1504 21.69589071 3.085479355
4408 21.68579676 3.271954388
2583 21.68055127 4.524550242
2077 21.67170631 4.543136891
2036 21.66438772 4.898812983
1178 21.65331963 4.401538343
3639 21.64419237 3.816461305
4270 21.63621983 4.551707436
5055 21.62782057 4.745034055
3807 21.62039496 4.817210707
4245 21.61590445 4.67510016
714 21.59075277 2.670359992
4553 21.5855391 4.856607338
3986 21.58387227 3.682497538
1277 21.57038595 3.119320275
3685 21.55638401 4.607028266
4397 21.54625512 5.136488661
4757 21.54307459 4.89210837
3035 21.54306103 4.967821152
5306 21.54258367 4.594438027
1943 21.52956653 4.851311554
522 21.52574939 4.994814467
5175 21.52537839 3.102782056
4309 21.50994471 3.08309742
4302 21.50853574 4.236725154
4215 21.50292316 4.865887049
2964 21.4956509 3.697963716
4793 21.49450508 4.245563169
4565 21.49083817 5.144127478
3274 21.49054901 5.093662984
3863 21.48137177 4.777152107
5215 21.47370977 3.22069172
2027 21.44760176 4.366770482
1994 21.43972987 4.750737103
3952 21.43484489 4.47236129
4041 21.41595345 2.967101762
877 21.41179428 4.787985601
3845 21.40848723 4.541872123
601 21.39685157 4.475333597
2039 21.39581163 3.379038545
4106 21.39537829 4.160526143
4630 21.38437352 4.463742004
5121 21.37983429 4.293707194
3643 21.37650303 5.034886134
3933 21.36985611 3.272435484
5130 21.36698952 4.871276019
5068 21.35245502 4.310251515
1599 21.34901055 5.176705717
1027 21.3489523 2.997870771
1446 21.34780443 4.67755418
3344 21.3475913 5.126309454
2662 21.34006106 4.98250531
4224 21.33050119 3.994020769
4547 21.33037418 3.203428486
3814 21.32236762 4.283181654
3082 21.32116122 4.848600871
3991 21.31785457 3.630087434
4285 21.31468156 3.103773405
5156 21.31423375 3.348861433
2676 21.30814954 3.194380719
4155 21.3078797 4.633550192
5017 21.30096465 3.305793609
4912 21.2978298 4.548408817
3630 21.29754138 4.998866814
4466 21.29740372 4.785732087
4533 21.29386296 3.446496501
4354 21.28892726 4.536608983
395 21.28336399 3.848249324
2151 21.28068234 4.500279208
3359 21.27758968 4.841163321
1529 21.27642641 3.243283798
3180 21.25992397 4.462610375
753 21.25913395 3.460586959
5118 21.25593764 3.970630138
3488 21.24335832 4.365234818
1763 21.24332196 4.654309297
3295 21.22865208 4.388670725
1414 21.21703214 4.99481088
1474 21.21549678 3.448261672
4123 21.20780838 4.577309113
3703 21.20091858 4.646266974
2854 21.1913235 4.194972631
4663 21.18941761 4.851033915
5117 21.18820596 4.21356447
5187 21.18596079 4.272125718
4817 21.17692475 4.649660578
5273 21.173849 4.89670621
4664 21.17291107 4.819354549
4010 21.17124083 2.747217862
2360 21.16075999 4.585532052
5218 21.15090927 4.866989684
4304 21.14652998 3.37115834
2982 21.13994161 5.066436853
4141 21.13915291 4.821611787
3329 21.13685735 5.069576529
4581 21.12759357 4.675741163
4611 21.11521746 4.800812629
3878 21.1138723 4.501611382
5146 21.08428644 4.211355099
3354 21.07704768 2.685275302
5335 21.06012328 4.080701449
4018 21.05931428 4.919112335
4149 21.05717351 4.814004355
854 21.04734925 3.41538174
3799 21.03083937 4.305360695
4804 21.01942597 4.490884223
3225 21.01416542 4.217633583
619 20.99989342 4.927975014
2143 20.98466196 4.970786085
5246 20.96967684 4.104676424
4859 20.96489022 4.257149195
4648 20.94640939 4.833549723
156 20.94555147 4.652610035
5226 20.93968662 3.115619483
4316 20.93896763 4.830446444
4186 20.93589489 4.032210787
3062 20.92655915 4.563959792
3957 20.92041376 4.675008504
5007 20.90423664 4.13060361
3927 20.89983223 3.240823443
5225 20.85090698 3.143680396
4776 20.8479108 3.262063496
896 20.84758177 4.50114434
4043 20.82319454 2.930768404
115 20.81697907 3.218887154
5249 20.79850177 4.73208868
5006 20.79493825 2.717846667
31 20.79385077 4.831878256
3379 20.78643169 4.625677167
3544 20.7687236 4.765015272
173 20.74532193 2.947850502
5092 20.74256545 4.584668152
4738 20.70476614 4.420157425
3839 20.69743558 4.875044566
2357 20.68302233 4.782461347
4634 20.66783734 4.153589823
3951 20.65718599 3.992471403
4702 20.62515841 4.320257542
3749 20.62179377 4.553489848
4032 20.59169305 4.702833828
4246 20.57083704 4.307840417
2985 20.55021355 5.065814374
3856 20.54097072 4.709572569
4470 20.52399516 4.411489758
3176 20.52208439 3.283907754
4015 20.51887181 4.362644621
4989 20.508978 4.494458826
3215 20.50159523 4.784623063
1765 20.48261734 4.390054158
974 20.48249769 4.775726924
136 20.48068622 4.512964347
2221 20.46907134 5.016719803
4670 20.44329369 4.941664942
352 20.43943739 4.527657173
4966 20.43497007 4.737671775
4336 20.43318843 4.913644463
5278 20.40047309 4.919373702
1660 20.37517344 5.142832076
4105 20.36905545 3.828472397
4920 20.32738779 4.114863254
1246 20.31108094 4.819283221
4711 20.29370333 4.358252638
4728 20.29300677 4.740396312
5169 20.29277159 4.858911358
5259 20.28312638 4.574089323
5067 20.25958097 3.927867108
4360 20.25813998 4.776763064
2477 20.25795419 4.579814405
4846 20.24459106 4.661480764
5329 20.19141231 4.190114638
4714 20.17125084 4.867346437
3110 20.16375122 4.418730565
2922 20.14115757 3.106269793
3023 20.13862869 4.809368998
3381 20.12996489 3.477462198
4195 20.12314932 4.932339995
3341 20.10482871 4.473821764
3345 20.07522164 5.004159653
2503 20.07143388 5.075185528
3202 20.05791935 4.941794465
2544 20.04573078 4.775037264
4798 20.03497174 4.267007361
2989 20.01757204 4.967193993
3283 20.01572011 4.818010449
4960 19.96001892 4.854257242
3502 19.94178834 4.902109685
3752 19.92516463 5.06167185
4595 19.89406327 3.090992729
3224 19.87622243 5.062065581
4517 19.87436638 4.159004906
3091 19.75414816 5.168307401
5217 19.74444312 3.22244583
5250 19.73695704 4.471346384
1604 19.72607703 5.058885127
1123 19.67365046 4.713400863
5009 19.6402823 4.807841978
4396 19.61667916 4.697495313
4095 19.61262627 4.248177689
5269 19.60259206 4.786578845
1254 19.60082642 4.009390825
2261 19.55763904 4.941666138
2389 19.51428737 4.476945134
4996 19.48763311 4.566497254
5202 19.40411802 5.086210806
1872 19.36393635 5.096624757
4019 19.35554454 4.526344484
4684 19.30942417 5.02132926
4087 19.27447363 4.962870211
4126 19.19054108 5.046593161
2952 19.16971411 5.224935251
5075 19.16296996 4.892465659
4142 19.04934212 4.793109701
2667 19.04587466 4.860844453
2382 18.9203 4.571932289
2998 18.9054508 4.905657549
5099 18.88527598 4.554081496
615 18.82720953 4.804459265
3740 18.81277744 5.177990025
4093 18.68793669 4.447856246
4144 18.61942464 4.442683844
2879 18.21441237 4.961698142
4671 18.16199962 4.753591698
5049 17.88317913 4.650154039
3258 17.50461155 5.122638833
3648 17.49845565 4.678098732
5205 17.26544666 4.530085423
996 17.24465971 4.990684298



Python Code:

from trueskill import *
import csv
import sys

filename = str(sys.argv[1])



nteams = 10000
teams = []
nmatches = []
for count in xrange(nteams):
x = Rating()
n = 0
teams.append(x)
nmatches.append(n)

for row in open(filename, 'rb'):
r1, r2, r3, b1, b2, b3, rs, bs = row.split('\t')
rt = [teams[int(r1)], teams[int(r2)], teams[int(r3)]]
bt = [teams[int(b1)], teams[int(b2)], teams[int(b3)]]
if (int(rs) > int(bs)):
(teams[int(r1)], teams[int(r2)], teams[int(r3)]), (teams[int(b1)], teams[int(b2)], teams[int(b3)]) = rate([rt, bt], ranks=[0, 1])
if (int(rs) < int(bs)):
(teams[int(r1)], teams[int(r2)], teams[int(r3)]), (teams[int(b1)], teams[int(b2)], teams[int(b3)]) = rate([rt, bt], ranks=[1, 0])
else:
(teams[int(r1)], teams[int(r2)], teams[int(r3)]), (teams[int(b1)], teams[int(b2)], teams[int(b3)]) = rate([rt, bt], ranks=[0, 0])

nmatches[int(r1)] += 1
nmatches[int(r2)] += 1
nmatches[int(r3)] += 1
nmatches[int(b1)] += 1
nmatches[int(b2)] += 1
nmatches[int(b3)] += 1

rankedteams = []
teamnums = []

for count in range(nteams):
if nmatches[count] > 0:
rankedteams.append(teams[count])
teamnums.append(count)

for count in range(len(teamnums)):
print teamnums[count], rankedteams[count].mu, rankedteams[count].sigma


Accepts a filename as an argument. The file should be a tab delimited file of red1 red2 red3 blue1 blue2 blue3 redscore bluescore

[connor.worley]

Very interesting results, thanks for reviving this!

[Michael Hill]

I think there are quite a few biases still because of the way we do regionals/districts. Teams that only compete in a week 1 or 2 regional and do well (but not well enough to go to championship), will be rated artificially high because the only teams they're compared against are the other teams at the regional who still have the default rating. There is no opportunity to mark them down later on if they don't compete again (but other teams from that regional do). What would be possible is to use results from prior years as well, but shift them toward the mean at the beginning of each season. This would account for good teams tend to consistently be good teams (to paraphrase Jim Zondag). This sort of thing is done in the Elo rankings of NFL teams by Nate Silver (http://fivethirtyeight.com/datalab/introducing-nfl-elo-ratings/) There are some other things we can take away from Silver as well. For example, we could possibly modify the algorithm to weigh eliminations more than the qualifications, something that's also done in the World Football Elo Ratings. See this wiki article: http://en.wikipedia.org/wiki/World_Football_Elo_Ratings#Basic_calculation_princ iples Basically, the K value (the factor that determins how much to change a participating team's elo rating) is modified based on how important a match is.

[Ether]

The file should be a tab delimited file of red1 red2 red3 blue1 blue2 blue3 redscore bluescore

Would you please post a ZIP of your input data file?

[Michael Hill]

Would you please post a ZIP of your input data file?




It's the same one I PMed you a couple weeks ago, but sure.

https://dl.dropboxusercontent.com/u/5193107/scores.zip

You'll need to install the trueskill package ('pip install trueskill') to use it.

[Michael Hill]

So I decided to take this data a little bit further. What I did was to take all of these calculations and run them through the data this year and try to predict matches. For this, I also included a modified Elo system that has diminishing returns for large margins of victory (Calling this Elo Mod). I got some rather surprising results.

My baseline was just using OPR for predicting match outcomes, it was able to predict about 77.1% of the matches this year. This was calculated by adding up the OPRs of each alliance and comparing with the result of the match. TrueSkill was able to predict 79.0% of the matches, a pretty good improvement. I need to develop the prediction model a bit better because it currently doesn't take into account the standard deviation as a measure of certainty. The modified Elo system was able to predict 79.5% of matches, an improvement over TrueSkill. The baseline, unadulterated Elo system as used in this thread was able to predict a whopping 81.4% of matches, by far the best out of any of these models. There is still room for improvement with the TrueSkill and Modified Elo. With the modified Elo, there are some constants that can be tuned for better results. But overall, the results are somewhat interesting. It seems that no matter the ranking model used, about 1 in 5 qualification matches will result in an upset.

Here is the spreadsheet I used: https://dl.dropboxusercontent.com/u/5193107/Elo%20Trueskill.xlsx

[Ether]

What I did was to take all of these calculations and run them through the data this year and try to predict matches.

Since you used the word "predict", is it safe to say that what you did was to use the data from weeks 1 through n-1 to predict the matches in week n?

[Michael Hill]

Since you used the word "predict", is it safe to say that what you did was to use the data from weeks 1 through n-1 to predict the matches in week n?




No, I guess I should have used a better word than "predict". More like "postdict". I went into it "knowing" an Elo/TrueSkill rating and tested against the data that was used to calculate it. Of course we won't have all the data to calculate the final Elo ratings during the season. My next step is to calculate all the ratings as if it is just before championships and then see how each does with "predicting" championship matches. My SWAG is that the success rate in predictions using the "postdicted" matches is a ceiling for how good we can hope for the predictions to be.

[Ether]

My baseline was just using OPR for predicting match outcomes, it was able to predict about 77.1% of the matches this year. This was calculated by adding up the OPRs of each alliance and comparing with the result of the match. TrueSkill was able to predict 79.0% of the matches, a pretty good improvement. I need to develop the prediction model a bit better because it currently doesn't take into account the standard deviation as a measure of certainty. The modified Elo system was able to predict 79.5% of matches, an improvement over TrueSkill. The baseline, unadulterated Elo system as used in this thread was able to predict a whopping 81.4% of matches, by far the best out of any of these models.

Try calculating "OPR" using min L1 norm of residuals (LAD) instead of min L2 norm (least squares), and see how that compares.

[Michael Hill]

I've tried the L1 optimization problem, but l1-magic is giving me fits. For some reason, it blows up after 10 iterations.

However, I have done the analysis that I wanted. I calculated all the rating systems using data from events prior to CMP, then I used that data to predict CMP matches.

In summary:
OPR: 72.46% Correct
TrueSkill: 69.31% Correct
Elo: 72.90% Correct
Elo Mod: 71.71% Correct

TL;DR: We're okay, but not great at predicting matches. OPR is okay at it, but Elo is better.

I'm still somewhat surprised that Elo is slightly better.

Updated Spreadsheet: https://dl.dropboxusercontent.com/u/5193107/Elo%20Trueskill%202.xlsx

[Ether]

I've tried the L1 optimization problem, but l1-magic is giving me fits. For some reason, it blows up after 10 iterations.

What solver are you using ?

[Ether]

In summary:
OPR: 72.46% Correct
TrueSkill: 69.31% Correct
Elo: 72.90% Correct
Elo Mod: 71.71% Correct

Try CCWM.

[Michael Hill]

What solver are you using ?




I'm using the l1eq_pd.m function in the L1-Magic library (http://users.ece.gatech.edu/~justin/l1magic/)

Try CCWM.



CCWM: 71.41%

[Ether]

I'm using the l1eq_pd.m function

That's the wrong solver.


% l1eq_pd.m
%
% Solve
% min_x ||x||_1 s.t. Ax = b

Firstly, you cannot find a min L1 norm vector x such that Ax=b because there is no vector x such that Ax=b, since the system is overdetermined.

Secondly, what you want to find is the min L1 norm of the residuals, not of the solution vector itself.

For the set of overdetermined linear equations Ax ≈ b, x is the solution vector. The residuals are b-Ax. So you want find a solution vector x which minimizes the L1 norm of b-Ax.

[Ether]

Attached is a comparison of b-Ax residuals for L2 and L1 OPR.

Alliance scores computed from L1 OPR are within +/-10 points of the actual scores 33.5% of the time.

Alliance scores computed from L2 OPR are within +/-10 points of the actual scores only 22.4% of the time.

It is on that basis that I postulate that L1 OPR might be a better predictor of match outcome.

Cannot add attachments to threads associated with papers. Brandon: can you please change this setting to allow attachments? Thank you.

[Ether]

My baseline was just using OPR for predicting match outcomes, it was able to predict about 77.1% of the matches this year.

I just ran the OPR numbers using the data you linked (http://www.chiefdelphi.com/forums/showpost.php?p=1403503&postcount=17) in your earlier post. I came up with 6919 of 8921 matches correctly "postdicted", or 77.56%

Our numbers are very close, but I had expected them to be identical.

Here's a link (http://ether.comeze.com/FRC/OPR4812/) to an XLS spreadsheet.

[Michael Hill]

I just ran the OPR numbers using the data you linked (http://www.chiefdelphi.com/forums/showpost.php?p=1403503&postcount=17) in your earlier post. I came up with 6919 of 8921 matches correctly "postdicted", or 77.56%

Our numbers are very close, but I had expected them to be identical.

Here's a link (http://ether.comeze.com/FRC/OPR4812/) to an XLS spreadsheet.





The differences are how we're calling tied matches. The way you're calculating if OPR "correctly" predicted falls apart when the actual result of the match was a tie. Almost always, the OPR will not predict a tie.

[Michael Hill]

Attached is a comparison of b-Ax residuals for L2 and L1 OPR.

Alliance scores computed from L1 OPR are within +/-10 points of the actual scores 33.5% of the time.

Alliance scores computed from L2 OPR are within +/-10 points of the actual scores only 22.4% of the time.

It is on that basis that I postulate that L1 OPR might be a better predictor of match outcome.

Cannot add attachments to threads associated with papers. Brandon: can you please change this setting to allow attachments? Thank you.




Ya, I'll admit I don't know what I'm doing with the L1 stuff. It's a new concept to me, so I just did some quick googling and thought I found a quick solution.

[Ether]

The differences are how we're calling tied matches.

77.75% correct (discarding ties)

77.11% correct (counting ties as incorrect)

77.93% correct (counting ties as correct)

[Michael Hill]

77.75% correct (discarding ties)

77.11% correct (counting ties as incorrect)

77.93% correct (counting ties as correct)




It looks like we're in agreement when counting ties as incorrect.

[Ether]

...the L1 stuff [is] a new concept to me...

You have an overdetermined linear system

Ax ≈ b,

where A is the (binary) design matrix of alliances, b is a column vector of alliance scores, and x is what you are trying to find: a column vector of team "OPR" scores.

There is no exact solution for x, since the system is overdetermined. So the idea is to find the "best" solution (in some sense of the word "best").

Notice that the left-hand side (Ax) is a column vector of alliance scores computed from whatever solution x you come up with.

The residuals are b-Ax: a column vector of the differences between the actual alliance scores (b) and the computed alliance scores (Ax).

Looking at it that way, it becomes clear that what you are trying to do is find a solution x which minimizes the residuals (in some sense of the word "minimize").

The most common way to do this is to find x which minimizes the L2 norm of the residuals. The L2 norm of a vector is the square root of the sum of the squares of the vector's elements. The L2 norm solution is also known as the "least squares" solution (for obvious reasons).

It turns out that finding the x which minimizes the L2 norm of b-Ax is computationally straightforward.

In Octave, it's one line of code: x = A\b. The backslash in this context is known as "left division". The syntax is simple, but under the hood there's a lot going on.

For the Ax ≈ b overdetermined linear systems were are dealing with in FRC to compute OPR scores, it turns out that there is a computationally faster way to compute the least squares solution for x. Here's how:

Multiply both sides of Ax ≈ b by the transpose of A to get A'Ax = A'b, or Nx =d where N=A'A and d = A'b.

Nx =d is known as the system of "Normal Equations", and its solution x = N\d gives the same answer as A\b (within rounding error) and is faster to compute.

But "least squares" (min L2 norm of residuals) is not the only possible "best fit" solution to the overdetermined system Ax ≈ b.

For example, there's the "Least Absolute Deviations (LAD)" solution (min L1 norm of residuals). The L1 norm of a vector is the sum of the absolute values of the vector's elements.

Finding an LAD solution for Ax ≈ b is more computationally intensive than least squares.

Perhaps the best way to proceed is to convert the problem to a "Linear Program" (LP) and then use one of the many LP solvers.

For example, here's the AMPL code I used to compute the LAD OPR for your data:

param m;
param n;

set I := {1..m};
set J := {1..n};

param A{I,J};
param b{I};

var x{J};
var t{I} >= 0;

minimize sum_dev:
sum {i in I} t[i];

subject to lower_bound {i in I}:
-t[i] <= b[i] - sum {j in J} A[i,j]*x[j];

subject to upper_bound {i in I}:
b[i] - sum {j in J} A[i,j]*x[j] <= t[i];