Re: ECAC Projected Standings 2012-13
By a whole six tenths of a point (in the most recent unposted midweek update)!
By a whole six tenths of a point (in the most recent unposted midweek update)!
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ECAC Projected Standings 2012-13
- Thread starter lugnut92
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FlagDUDE08
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Re: ECAC Projected Standings 2012-13
What? No decimal points? None of these predictions contains a significant figure. Scientifically accurate (impressive) measurements require decimal points, and the more decimal places and zeros the better, regardless of the validity of the data base.
What? No decimal points? None of these predictions contains a significant figure. Scientifically accurate (impressive) measurements require decimal points, and the more decimal places and zeros the better, regardless of the validity of the data base.
FlagDUDE08
Banned
Re: ECAC Projected Standings 2012-13
The next time you calculate the area of a circle, use 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612
The next time you calculate the area of a circle, use 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612
Re: ECAC Projected Standings 2012-13
1. Quinnipiac (34.5555020182225)
2. Dartmouth (33.4459855498100)
3. Yale (26.5138029601541)
4. Cornell (25.5665853859834)
-----
5. Union (24.0336346151021)
6. Colgate (21.8964717311746)
7. Harvard (20.3846365157740)
8. Clarkson (16.6299551022060)
-----
9. Princeton (16.0335304974869)
10. St. Lawrence (15.6265247209489)
11. RPI (14.9573744325469)
12. Brown (14.3559964705906)
(Rounding error= 0.0000000000000)
Better? I round because teams can only have integral point totals. I give extra digits when it would otherwise appear to be a tie.
1. Quinnipiac (34.5555020182225)
2. Dartmouth (33.4459855498100)
3. Yale (26.5138029601541)
4. Cornell (25.5665853859834)
-----
5. Union (24.0336346151021)
6. Colgate (21.8964717311746)
7. Harvard (20.3846365157740)
8. Clarkson (16.6299551022060)
-----
9. Princeton (16.0335304974869)
10. St. Lawrence (15.6265247209489)
11. RPI (14.9573744325469)
12. Brown (14.3559964705906)
(Rounding error= 0.0000000000000)
Better? I round because teams can only have integral point totals. I give extra digits when it would otherwise appear to be a tie.
Re: ECAC Projected Standings 2012-13
Only two OOC games were played this weekend, but the North Country's losses dropped them both a good deal. Here are your first ECAC intermission standings:
1. Quinnipiac (35)
2. Dartmouth (34)
3. Yale (26.4)
4. Cornell (25.5)
-----
5. Union (24.6)
6. Colgate (22)
7. Harvard (21)
8. Princeton (16.0)
-----
9. RPI (15.8)
10. Clarkson (15.4)
11. Brown (14.5)
12. St. Lawrence (14.2)
(Rounding error: 0)
St. Lawrence's epic descent has reached bottom after their loss to Vermont. Clarkson's loss also put them back on the road. 8th and 12th are separated by less than two points, so we could be in for a wild fight for the last home spot. That said, a middle tier seems to have emerged from 3rd to 7th.
Only two OOC games were played this weekend, but the North Country's losses dropped them both a good deal. Here are your first ECAC intermission standings:
1. Quinnipiac (35)
2. Dartmouth (34)
3. Yale (26.4)
4. Cornell (25.5)
-----
5. Union (24.6)
6. Colgate (22)
7. Harvard (21)
8. Princeton (16.0)
-----
9. RPI (15.8)
10. Clarkson (15.4)
11. Brown (14.5)
12. St. Lawrence (14.2)
(Rounding error: 0)
St. Lawrence's epic descent has reached bottom after their loss to Vermont. Clarkson's loss also put them back on the road. 8th and 12th are separated by less than two points, so we could be in for a wild fight for the last home spot. That said, a middle tier seems to have emerged from 3rd to 7th.
Re: ECAC Projected Standings 2012-13
No ECAC teams played this week, but every team that did play has played ECAC teams this season, so KRACH did change. Intermission week 2 standings:
1. Quinnipiac (35)
2. Dartmouth (34)
3. Yale (26)
4. Cornell (25.5)
-----
5. Union (24.6)
6. Colgate (22)
7. Harvard (21)
8. Princeton (16.01)
-----
9. RPI (16.00)
10. Clarkson (15)
11. Brown (14.4)
12. St. Lawrence (14.1)
(Rounding error: -1)
Not going to draw any meaning from these.
No ECAC teams played this week, but every team that did play has played ECAC teams this season, so KRACH did change. Intermission week 2 standings:
1. Quinnipiac (35)
2. Dartmouth (34)
3. Yale (26)
4. Cornell (25.5)
-----
5. Union (24.6)
6. Colgate (22)
7. Harvard (21)
8. Princeton (16.01)
-----
9. RPI (16.00)
10. Clarkson (15)
11. Brown (14.4)
12. St. Lawrence (14.1)
(Rounding error: -1)
Not going to draw any meaning from these.
Re: ECAC Projected Standings 2012-13
Now that we're through most of the holiday games, here are your final "intermission" projected standings:
1. Quinnipiac (36)
2. Dartmouth (35)
3. Yale (24.9)
4. Union (24.6)
-----
5. Cornell (24.5)
6. Colgate (22)
7. Harvard (19)
8. RPI (17)
-----
9. Princeton (15.8)
10. Clarkson (15.5)
11. St. Lawrence (14.9)
12. Brown (14.7)
(Rounding error: 0)
The race for the last bye is very tight (3-5: 0.4 pts), as is the bottom four (1.1 pts). RPI has finally jumped into home ice after their split with SCSU and Cornell has dropped out of a bye, though just barely. The three tiers I mentioned a few weeks ago have blurred slightly on the low end; Quinnipiac and Dartmouth remain miles ahead, but the lower two tiers are more evenly spread now. Quinnipiac has the toughest remaining league schedule, while RPI has the easiest. This week's match-up between the top two could serve to decide who comes away with the Cleary Cup come March.
Now that we're through most of the holiday games, here are your final "intermission" projected standings:
1. Quinnipiac (36)
2. Dartmouth (35)
3. Yale (24.9)
4. Union (24.6)
-----
5. Cornell (24.5)
6. Colgate (22)
7. Harvard (19)
8. RPI (17)
-----
9. Princeton (15.8)
10. Clarkson (15.5)
11. St. Lawrence (14.9)
12. Brown (14.7)
(Rounding error: 0)
The race for the last bye is very tight (3-5: 0.4 pts), as is the bottom four (1.1 pts). RPI has finally jumped into home ice after their split with SCSU and Cornell has dropped out of a bye, though just barely. The three tiers I mentioned a few weeks ago have blurred slightly on the low end; Quinnipiac and Dartmouth remain miles ahead, but the lower two tiers are more evenly spread now. Quinnipiac has the toughest remaining league schedule, while RPI has the easiest. This week's match-up between the top two could serve to decide who comes away with the Cleary Cup come March.
Re: ECAC Projected Standings 2012-13
The second half (actually more like middle third) of the ECAC schedule got underway this weekend with the Harvard-Dartmouth travel pair facing the Princeton-Quinnipiac partners. In addition, plenty of teams had OOC contests. Here are your updated standings:
1. Quinnipiac (39)
2. Dartmouth (28)
3. Yale (25)
4. Cornell (24)
-----
5. Union (23.274)
6. Colgate (23.267)
7. Princeton (22)
8. RPI (17)
-----
9. Harvard (16.4)
10. Clarkson (16.0)
11. St. Lawrence (15)
12. Brown (14)
(Rounding error: -2)
While Dartmouth's pair of losses didn't knock them out of second, their expected point total dropped significantly, leaving Quinnipiac all alone waaaay out in front. Harvard's pair of losses and Princeton's pair of wins led to their swapped positions. Princeton has effectively joined the middle tier, which now spreads from 2nd(ish) to 7th. Also in this tier, Cornell and Union each lost both of their games this weekend, but Cornell's KRACH suffered less and bumped them back into a bye. Speaking of Union, the closeness between Union and Colgate (0.007 pts) tempts me to call it an out and out tie (Colgate is expected to win that series 2.3-1.7), but thousandths of a point are still points. The toughest and easiest remaining league schedules belong to Yale and Dartmouth, respectively.
The second half (actually more like middle third) of the ECAC schedule got underway this weekend with the Harvard-Dartmouth travel pair facing the Princeton-Quinnipiac partners. In addition, plenty of teams had OOC contests. Here are your updated standings:
1. Quinnipiac (39)
2. Dartmouth (28)
3. Yale (25)
4. Cornell (24)
-----
5. Union (23.274)
6. Colgate (23.267)
7. Princeton (22)
8. RPI (17)
-----
9. Harvard (16.4)
10. Clarkson (16.0)
11. St. Lawrence (15)
12. Brown (14)
(Rounding error: -2)
While Dartmouth's pair of losses didn't knock them out of second, their expected point total dropped significantly, leaving Quinnipiac all alone waaaay out in front. Harvard's pair of losses and Princeton's pair of wins led to their swapped positions. Princeton has effectively joined the middle tier, which now spreads from 2nd(ish) to 7th. Also in this tier, Cornell and Union each lost both of their games this weekend, but Cornell's KRACH suffered less and bumped them back into a bye. Speaking of Union, the closeness between Union and Colgate (0.007 pts) tempts me to call it an out and out tie (Colgate is expected to win that series 2.3-1.7), but thousandths of a point are still points. The toughest and easiest remaining league schedules belong to Yale and Dartmouth, respectively.
Re: ECAC Projected Standings 2012-13
Out of curiosity, by what metric does Yale have the toughest remaining schedule? Shouldn't Brown have the toughest? Yale and Brown have the same remaining ECAC contests, with the exception of one head-to-head, and I would guess that Yale ranks higher than Brown. Does this consider that the head-to-head will be a road game for Yale?
Out of curiosity, by what metric does Yale have the toughest remaining schedule? Shouldn't Brown have the toughest? Yale and Brown have the same remaining ECAC contests, with the exception of one head-to-head, and I would guess that Yale ranks higher than Brown. Does this consider that the head-to-head will be a road game for Yale?
Re: ECAC Projected Standings 2012-13
Home-ice advantage is not considered. Based on how many points a team is expected to take from here in, we can figure out their average points per game. Then I solve for the "average" opponent's KRACH based on the following equation: points expected = 2*KRACH/(KRACH+Opponent KRACH). Yale's average opponent KRACH is 125, and Brown's is 121.9.
It is a little confusing for me that they are so low, given that the average KRACH of Brown's remaining opponents (for example) is around 166. Using this KRACH, however, shorts Brown about 2 points. If anyone could suggest a reason for the difference between "average oppenent" KRACH and opponents' average KRACH, I'm all ears.
Home-ice advantage is not considered. Based on how many points a team is expected to take from here in, we can figure out their average points per game. Then I solve for the "average" opponent's KRACH based on the following equation: points expected = 2*KRACH/(KRACH+Opponent KRACH). Yale's average opponent KRACH is 125, and Brown's is 121.9.
It is a little confusing for me that they are so low, given that the average KRACH of Brown's remaining opponents (for example) is around 166. Using this KRACH, however, shorts Brown about 2 points. If anyone could suggest a reason for the difference between "average oppenent" KRACH and opponents' average KRACH, I'm all ears.
Re: ECAC Projected Standings 2012-13
Now, Brown (with their 64.97 KRACH) is expected to take 1.16 points against Tech and 0.45 points against Dartmouth. Yale and its 189.39 Rating is expected to take 1.60 and 0.92 points, respectively in their two "remaining games."
That means that Brown is expected to get 1.61 points, or 0.81 wins, in their two remaining games. If Brown was facing one team twice and were expected to get 0.81 wins over the course of those two games, they would be playing a team with a rating of 96.33. Hence, their remaining strength of schedule is 96.33. Yale is expected to get 2.52 points, or 1.26 wins over their two remaining games. Since the Elis' KRACH is 189.39, their mythical opponent that they would play twice and earn 1.26 wins from would have a rating of 111.21.
If you did a straight average of the opponents' KRACH, you'd come up with 270.26, but then you're not giving Brown and Yale their due. Brown would only earn 0.78 points (points, not wins) over two games against the mythical 270.26 rated team and Yale would only earn 1.65 points over their two games. Both teams are expected to do better than that.
Now you understand why the values are different. However, I'm struggling trying to find a straightforward way of explaining why Yale's strength of schedule is higher despite the fact that they're playing the same teams.
At the very least, you can take heart in the fact that you know that you are calculating strength of schedule properly.
Let's consider it this way. Say that Brown and Yale had only two games each remaining in their season, one against Clarkson and one against Dartmouth. (Side tangent: I wanted to use Clarkson and Quinnipiac, but they each have two games remaining against Qpac so that makes it more difficult, so I went with the ECAC team with the lowest and highest KRACHs that they only play once.)
Now, Brown (with their 64.97 KRACH) is expected to take 1.16 points against Tech and 0.45 points against Dartmouth. Yale and its 189.39 Rating is expected to take 1.60 and 0.92 points, respectively in their two "remaining games."
That means that Brown is expected to get 1.61 points, or 0.81 wins, in their two remaining games. If Brown was facing one team twice and were expected to get 0.81 wins over the course of those two games, they would be playing a team with a rating of 96.33. Hence, their remaining strength of schedule is 96.33. Yale is expected to get 2.52 points, or 1.26 wins over their two remaining games. Since the Elis' KRACH is 189.39, their mythical opponent that they would play twice and earn 1.26 wins from would have a rating of 111.21.
If you did a straight average of the opponents' KRACH, you'd come up with 270.26, but then you're not giving Brown and Yale their due. Brown would only earn 0.78 points (points, not wins) over two games against the mythical 270.26 rated team and Yale would only earn 1.65 points over their two games. Both teams are expected to do better than that.
Now you understand why the values are different. However, I'm struggling trying to find a straightforward way of explaining why Yale's strength of schedule is higher despite the fact that they're playing the same teams.
At the very least, you can take heart in the fact that you know that you are calculating strength of schedule properly.
Re: ECAC Projected Standings 2012-13
The answers are different because you can't add KRACHes (which you're doing when you calculate the opponents' average KRACH) - KRACH is always a ratio, not a difference. If A's rating is 150 and B's rating is 100, A is not "50 points better than B." Instead you have to always think in ratios, "A is 50% better than B", which tells you that A will win 3 out of 5 games from B (150/250= 3/5 = .600). Now throw a 3rd team in to the mix, C at a rating of 50, so that the "difference" (which you should ignore) between B and C is the same 50 points. Naively, you might think that B's chances over C are the same as A's chances over B, since both pairs are 50 points apart. In fact, B should take 2 out of 3 games (100/150 = 2/3 = .667). In other words, B is "more better" than C than A is better than B.
Forgetting the numbers for a moment, if we have 2 pairs of travel partners playing each other (A & B vs. C & D), then A would be expected to get: 2*( A/(A+C) + A/(A+D)) points on the weekend. If you averaged C's and D's ratings and played that opponent twice, you'd get 2*(A/(A + (C+D)/2) + A/(A+(C+D)/2) ) = 2*(4A/(2A + C +D), which is quite obviously a different answer (unless C happens to equal D). To calculate the mythical opponent's KRACH who would give you the correct expected points (call their rating X), you'd start from A/(A+C) + A/(A+D) = 2*A/(A+X). Solving for X gives:
X = 2(A+C)(A+D)/(2A+C+D) - A. Thinking about the partial derivative with respect to A (without actually calculating it), you've got an A^2 in the numerator and only A^1 in the denominator, so the mythical opponent's KRACH should increase as A increases.
I think the problem is simply in referring to the KRACH of these mythical opponents as "the strength of schedule," since that number will always be relative to the team in question. It is what it is (the KRACH of the opponent A would have to play N times to get the same number of points as if they played N different teams), but nothing more than that.
The answers are different because you can't add KRACHes (which you're doing when you calculate the opponents' average KRACH) - KRACH is always a ratio, not a difference. If A's rating is 150 and B's rating is 100, A is not "50 points better than B." Instead you have to always think in ratios, "A is 50% better than B", which tells you that A will win 3 out of 5 games from B (150/250= 3/5 = .600). Now throw a 3rd team in to the mix, C at a rating of 50, so that the "difference" (which you should ignore) between B and C is the same 50 points. Naively, you might think that B's chances over C are the same as A's chances over B, since both pairs are 50 points apart. In fact, B should take 2 out of 3 games (100/150 = 2/3 = .667). In other words, B is "more better" than C than A is better than B.
Forgetting the numbers for a moment, if we have 2 pairs of travel partners playing each other (A & B vs. C & D), then A would be expected to get: 2*( A/(A+C) + A/(A+D)) points on the weekend. If you averaged C's and D's ratings and played that opponent twice, you'd get 2*(A/(A + (C+D)/2) + A/(A+(C+D)/2) ) = 2*(4A/(2A + C +D), which is quite obviously a different answer (unless C happens to equal D). To calculate the mythical opponent's KRACH who would give you the correct expected points (call their rating X), you'd start from A/(A+C) + A/(A+D) = 2*A/(A+X). Solving for X gives:
X = 2(A+C)(A+D)/(2A+C+D) - A. Thinking about the partial derivative with respect to A (without actually calculating it), you've got an A^2 in the numerator and only A^1 in the denominator, so the mythical opponent's KRACH should increase as A increases.
I think the problem is simply in referring to the KRACH of these mythical opponents as "the strength of schedule," since that number will always be relative to the team in question. It is what it is (the KRACH of the opponent A would have to play N times to get the same number of points as if they played N different teams), but nothing more than that.
Re: ECAC Projected Standings 2012-13
Awesome explanation, LynahFan. The only quibble I have is that the explanation of KRACH on slack.net (and the generally accepted definition) uses that weighted average (that you've described so much more eloquently than I ever could) as the Strength of Schedule. It could probably more accurately be called the KRACH of a mythical opponent that Team A would play N times and have the record that they currently have, but that's a bit of a mouthful.
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Re: ECAC Projected Standings 2012-13
Thanks a ton to both LynahFan and burgie12. I thought that it may have something to do with being relative to a team's KRACH. I initially referred to the strength of schedule by mentioning a team's "average opponent" and I guess I'll go back to that. I wonder if there is some way to determine a strength of schedule that isn't relative to Team A's KRACH. Would geometric mean be a valid way to average KRACH since it involves multiplication rather than addition?
Thanks a ton to both LynahFan and burgie12. I thought that it may have something to do with being relative to a team's KRACH. I initially referred to the strength of schedule by mentioning a team's "average opponent" and I guess I'll go back to that. I wonder if there is some way to determine a strength of schedule that isn't relative to Team A's KRACH. Would geometric mean be a valid way to average KRACH since it involves multiplication rather than addition?
Re: ECAC Projected Standings 2012-13
For a "strength of schedule," I was initially thinking that you could take the mythical team's KRACH and divide it by the KRACH of the given team, but that devolves almost immediately into an inverse KRACH ranking.
As I'm sure you know (since you already use this formula), LynahFan's formula of "X = 2(A+C)(A+D)/(2A+C+D) - A" can also be described as "2 * (KRACH_A * Remaining_Games / Points_To_Be_Earned_In_Remaining_Games) - KRACH_A" (the "2 *" is included because there are two points earned per game). Taking KRACH_A out of that formula is just going to result in the same issue that I ran into in Paragraph 2, it's nearly an inverse KRACH rating.
I'm trying to experiment with other ranking systems, specifically the Mease Ranking that had been adapted to use ties that LakersFan uses on the Women's side of the forum (bolded for Ralph's benefit), in preparation for the Byes / Home-Ice thread that is going to start in the next couple of weeks (spoiler alert). Now, I just need to learn how to calculate it.
I was going to include that in my example this morning, too, but didn't for whatever reason. The basic answer is that it's better, but still isn't accurate. For example, the geometric mean of Clarkson and Dartmouth's KRACH is 102.45, so Brown would only be expected to get 1.55 points while Yale would get 2.60 in their two games against the two teams. Neither is correct (although it's a heck of a lot closer than using an arithmetic mean).
For a "strength of schedule," I was initially thinking that you could take the mythical team's KRACH and divide it by the KRACH of the given team, but that devolves almost immediately into an inverse KRACH ranking.
As I'm sure you know (since you already use this formula), LynahFan's formula of "X = 2(A+C)(A+D)/(2A+C+D) - A" can also be described as "2 * (KRACH_A * Remaining_Games / Points_To_Be_Earned_In_Remaining_Games) - KRACH_A" (the "2 *" is included because there are two points earned per game). Taking KRACH_A out of that formula is just going to result in the same issue that I ran into in Paragraph 2, it's nearly an inverse KRACH rating.
I'm trying to experiment with other ranking systems, specifically the Mease Ranking that had been adapted to use ties that LakersFan uses on the Women's side of the forum (bolded for Ralph's benefit), in preparation for the Byes / Home-Ice thread that is going to start in the next couple of weeks (spoiler alert). Now, I just need to learn how to calculate it.
Re: ECAC Projected Standings 2012-13
The way I actually calculate it is KRACH A*(Max points possible - expected points)/(expected points - current points). If you divide the KRACH A term out, you basically get (points expected to lose/points expected to win), which is identical to (1/expected point percentage for remaining games) - 1. That's still dependent on KRACH, so it doesn't help either.
The way I actually calculate it is KRACH A*(Max points possible - expected points)/(expected points - current points). If you divide the KRACH A term out, you basically get (points expected to lose/points expected to win), which is identical to (1/expected point percentage for remaining games) - 1. That's still dependent on KRACH, so it doesn't help either.