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topic this topic is 3 pages
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tangotiger |
posted July
25th, 2001 05:00 PM |
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All Star Member
Since: May 2000 Location:
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Given
that a team scores .500 runs/inning, how often will they
score 0,1,2 runs, etc?
Unlike hockey, baseball
does not follow a poisson distribution.
Anyway,
here's how you can figure it: RI = runs/inning =
.500 a = RI*RI*.73 = .183 f0 = RI/(RI+a) = .733
= frequency of innings with 0 runs dropRate = 1 -
.73*f0 = .465 = f2/f1 = f3/f2 = f4/f3 ...
fo +
f1 + f2 + ... + f20 = 1.00 solve for f1 (alot
easier to do with Excel, and in this case it equals
.143)
Anyway, here's the breakdown for the .500
example: 0 73.26% 1 14.30% 2 6.65% 3
3.09% 4 1.44% 5 0.67% 6 0.31% 7 0.14%
8 0.07% 9 0.03% 10 0.01% 11 0.01%
If you multiply the numbers out, you get exactly
.500. Now, I've tested this from .200 to 1.000, and it
works perfectly. I don't know why mathematically this
works out, but maybe I've stumbled upon the Tango
Distribution?
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Warren |
posted July
25th, 2001 05:17 PM |
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Senior
Member Member Since: Dec 1999 Location:
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Keith
Woolner looked at this a couple of years ago, if you're
interested...
http://www.baseballprospectus.com/news/20000304woolner.html
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CRS |
posted July
25th, 2001 06:09 PM |
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Senior
Member Member Since: Feb 2000 Location:
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How
about something like a teams record in 1-run games given
their RS/g and RA/g?
Extra credit if you can
figure out the likelihood of each element of the
score-matrix.
I don't know the answer, I just
think that would be an interesting question and you seem
good with numbers.
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tangotiger |
posted July
26th, 2001 10:07 AM |
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All Star Member
Since: May 2000 Location:
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Yes, I
think that was one of Keith's best work. No offense to
him, but I think my formula is a little easier to
remember and handle.
(I based my probability of
0 runs/inning based on my sim, and so my formula matches
that. However, seeing that he has actual data, I will
modify slightly my formula to match his data.)
I
was hoping that the math majors out there would have
saved me the trouble of turning r/inn into r/gp. Any
takers?
From that point, it is a simple matter
to get a win% probability matrix of rs/gp vs
ra/gp.
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Patriot |
posted July
31st, 2001 09:41 AM |
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All Star Member
Since: Jul 2000 Location: Ohio
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Another
extremely useful tool from Tango! I have fooled around
with this...please tell me if I'm wrong Tango, but based
on your example it seems as if: f1 =
(1-f0)*(1-Droprate)
If this is correct, than
this little test I did has some basis. James printed
1982 AL and 1983 NL data for x runs in innings. I think
Jarvis also has some available. Anyway, using this I
checked the predicted probability of scoring v. actual,
and the predicted probability of a big inning(James
defined as >=3 runs). Anyway, it SEEMED as if the
probability of scoring was a little low, and the
probability of a big inning was a little high. Please
note the word seemed as I didn't do a real careful
analysis of it. Also, I didn't have acutal innings
batted so I used G/9 to get RI which could hurt the
accuracy.
Also, Tango, what is the signifcance
of the .73 multiplier?
Anyway, great
work.
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bamadan |
posted July
31st, 2001 11:04 AM |
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All Star Member
Since: Aug 1999 Location:
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It has
been posited by Bill James and others that assuming two
teams of equal gross runs scored, the team that scores
runs 1 at a time rather than in bunches will have a
substantial benefit. As such, one run strategies such as
the sacrifice or steal may be less negative (or somewhat
positive at times) than previously thought.
Tango's theoretical work above, apparently
assumes random distribution of runs. Woolner's empirical
data isn't broken down into Mauchian and Weaverite
strategies. Does anyone know of the availability of a
study showing distribution of runs as impacted by
strategic decisions? Is the Bill James scenario a
meaningful construct or a hypothetical which exists only
in computer modeling?
As an aside, Jim Baker has
a daily e-mail newsletter which often contains
sabermetric commentary of that day's games. The
following piece on run distribution is lifted from the
7/30 edition:
quote:
ALL RUNS ARE NOT CREATED EQUALLY a guest
column by Jeff Fogle
Hi folks.
With
the dearth of games today, Jim thought it might be fun
to make public a topic that he and I have been
discussing on the phone and in emails the past few
weeks. On my end, it has gotten to the point that I
see almost any baseball issue through this particular
prism. It has changed my view of things to a
surprising degree.
We start with a premise put
forth by Bill James in one of the old Abstracts, and
add in a semi-related corollary from an article on
bunting in the BILL JAMES GUIDE TO BASEBALL MANAGERS,
(pages 132-133 specifically for those of you with the
book).
Part 1: Each run has value, but runs
one through five in a game are the most important and
each run thereafter has diminishing value. In other
words, all runs are important in a 4-3 game, but some
are superfluous in an 8-3 game. It has been more
than a decade since I read the exact article, but a
general point was made that a team scoring 5-5-5-5-5
each day would be a lot more successful than one
scoring 10-0-10-0-10-0. They would have the same
average but the first would win well over half their
games as long as they had an adequate pitching
staff, while the other would fail to even reach .500
because every shutout is a loss.
Part 2: From
the bunting article: Runs scored one at a time are
more valuable than runs scored in bunches. Because
pinpointing a way to measure the issue exactly is
difficult, the article put forth the notion that runs
scored one at a time could be worth as much as 11-24%,
or even 50% more than runs scored in bunches. James
then explained a study he ran which eventually
concluded that a team averaging 4.5 rpg scoring
one at a time would go 90-72 in a full season of games
facing a team that scored 4.5 rpg in three run bursts.
Same scoring average, but a clear head to head edge.
Those points offer the basic backdrop for the
following issues:
*THE SEATTLE MARINERS FIRST
HALF EXPLOSION: The Mariners had an unbelievable first
few months. And, the key was their ability to score
consistently. Up through July 3 (their first 82
games), they reached five runs in 72% of the time.
While most of baseball was struggling to score with
the new strike zone and fighting some cool weather in
the Northeast and Midwest, Seattle was on fire. No
team could truly go 5-5-5-5-5, of course. But Seattle
got close in the first half of the season, and proved
that such consistency equals dramatic success.
*IS ICHIRO SUZUKI AN MVP CANDIDATE?: A more
appropriate question back when he was hitting .350
(before he either did or did not hit a wall). But, at
the time of the amazing Seattle run, he was arguably
the lynch pin to their consistency. Around the
All-Star break he was the team leader in what you
could call OFFENSIVE BASES, which is the sum of Total
Bases, Walks, and Stolen Bases. (Those of you in
positions of influence, please feel free to push this
stat because it is amazing how it really emphasizes
the difference between hitters and non-hitters). The
team race was close. Because Seattle features a four-
to five-horse merry-go-round offense (with big
contributions from Bret Boone, Edgar Martinez, and Jon
Olerud too). But, as the lead horse of the most
amazing offensive run in years, Ichiro certainly could
have -- at that time -- been considered an MVP
candidate.
*WHY DO SUPERSTAR SLUGGERS FAIL TO
MAKE MORE OF AN IMPACT AFTER MARQUEE TRADES? Because
they do things in bunches, and runs scored in bunches
are less valuable than runs scored one at a time. At
the team level, 5-5-5-5-5 is better than
10-0-10-0-10-0. The daily production of the superstar
sluggers looks more like the latter. Imagine offensive
bases here instead of runs. Maybe the daily offensive
bases category looks something like 0-6-0-6-0-6
over the long haul. Another vote for Ichiro would
be that he was doing something more like 3-3-3-3-3-3
during the first half of the season. So, his
consistent production helped Seattle get to five runs
every night, while the guys who are most often
considered for MVP are pushing their teams closer to
the 10-0-10-0 framework by the very nature of their
production.
*WHY IS THERE SO MUCH COMPETITIVE
BALANCE IN BASEBALL? Because most teams are built
identically, with their version of the slugger in the
heart of the lineup. Those guys produce in random
bursts so the teams are scoring runs in random bursts
making it hard to careen out of the .420 to .580
range. Nobody is going to really go 10-0-10-0-10-0,
but there are a lot of 3-7-3-7-3-7 type teams. And
shuttling around the sluggers fails to change that, no
matter how much you pay the guy.
...
Okay, this is becoming a novel. The idea I
hope you will consider is that CONSISTENCY should be
the goal of offenses. And the marquee players who are
generally considered to be the offensive stars are
ironically anti-consistency, because they produce
in fits and starts. This locks their team on
day-to-day roller coasters that actually prevent the
teams from reaching greatness.
This is why
sluggers only end up being worth a few games a year.
And why a healthy Seattle found a relatively
slugger-less synergy that saw them standing at 61-21
on July 4th.
Not that I agree with everything
the author posits (OFFENSIVE BASES without context of
outs consumed is, at best, a half arsed stat) but
interesting none-the-less.
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tangotiger |
posted July
31st, 2001 11:44 AM |
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All Star Member
Since: May 2000 Location:
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Pat,
f1 = (1-f0)*(1-Droprate)
I didn't
even realize that it worked out that way! I'll have to
look into this.
As for the .73 thing, it's
trial-and-error. I "know" the chances that an inning
should have 0 runs, and started off with that. I
"realized" the relationship between R/I and % of innings
with 0 runs followed the relationship I mentioned.
The dropoff rate was also another thing that I
noticed from looking at high and low scoring teams
(based on my sim).
Just trial-and-error,
really....
I'll update it soon based on
Woolner's "actual" rates.
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tangotiger |
posted July
31st, 2001 01:06 PM |
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All Star Member
Since: May 2000 Location:
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Thanks
to Pat for that bit of insight, here is a functional
version of the runs/inn calculator (you'll need Excel):
http://www.geocities.com/tmasc/runsinn.xls
I've ran tests against Woolner's data. That
"control" value of .74 that I use also changes slightly
based on run environment. Can be anywhere from .73 to
.78 based on run environment. It does NOT alter the
results that much in any case. I suggest that you use
.74, and you'll get close enough data.
If
someone can explain the mathematics of why it all adds
up, I'd be thankful.
In any case, all you have
to do is edit cell E2, and you;ll get your distribution.
In my next release, I'll update it so that it
matches exactly with real-life, including a dynamic
value for the "control" value.
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tangotiger |
posted July
31st, 2001 08:21 PM |
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All Star Member
Since: May 2000 Location:
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Ok, I
input all of Woolner's real-life data, which is 20 YEARS
worth of R/I data.
The best "control" value to
use is .761. If anyone downloaded my file, make sure
that you use that value.
Using the data that
Woolner had, for the 3.0-3.5, 3.5-4.0, etc "classes",
this is the "control" value to use: .768 .772
.769 .757 .763 .751 .764
Overall, this works out to .761.
Here is
how it works out for the entire group: R/I Actual
Expected 0 73.05% 73.06% 1 14.81% 14.99% 2
6.76% 6.65% 3 3.05% 2.95% 4 1.37% 1.31% 5
0.57% 0.58% 6 0.24% 0.26% 7 0.09% 0.11% 8
0.04% 0.05% 9 0.02% 0.02% 10 0.01% 0.01%
Here's how Keith Woolner and I see the 3.0-3.5
class of teams:
R/I Actual Tango Woolner
0 78.13% 78.28% 79.00% 1 13.07% 12.95%
13.20% 2 5.22% 5.23% 4.90% 3 2.13% 2.11% 1.80%
4 0.93% 0.85% 0.70% 5 0.30% 0.34% 0.30% 6
0.12% 0.14% 0.10% 7 0.07% 0.06% 0.00% 8 0.01%
0.02% 0.00%
How about the 4.5-5.0 class?
R/I Actual Tango Woolner 0 71.95% 71.85%
72.70% 1 14.99% 15.40% 14.90% 2 7.14% 6.97%
6.80% 3 3.31% 3.16% 3.10% 4 1.51% 1.43% 1.40%
5 0.65% 0.65% 0.60% 6 0.27% 0.29% 0.30% 7
0.10% 0.13% 0.10% 8 0.05% 0.06% 0.10%
Finally, the 5.5-6.0 class?
R/I
Actual Tango Woolner 0 66.29% 66.39% 66.80% 1
16.29% 16.99% 16.50% 2 8.71% 8.40% 8.30% 3 4.85%
4.15% 4.20% 4 2.24% 2.05% 2.10% 5 0.86% 1.02%
1.10% 6 0.49% 0.50% 0.50% 7 0.20% 0.25% 0.30%
8 0.08% 0.12% 0.10%
Except for the 1 run
estimation, practically a clean sweep for my formula.
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tangotiger |
posted August
2nd, 2001 12:35 PM |
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All Star Member
Since: May 2000 Location:
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I am in
the middle of writing a program that will accept two
inputs: runs per inning of away team and home team.
Based on the run distribution formula I have presented,
it will then run through a million simulations to
calculate the win% for every pair of team-inputs.
(This simulator will be much more accurate than
my other one, because I am already starting off with the
run distribution by inning. It will also be alot faster
because I have less things to "simulate".)
I
will make the executable available on my website
sometime within the next few days. This should answer,
once and for all, the relationship between runs and
wins.
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David Smyth |
posted August
2nd, 2001 09:19 PM |
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All Star Member
Since: Dec 1999 Location: Lake Vostok
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What
about different types of offenses? What would the
distribution by inning be for am 80's Cardinal team,
averaging .5 run per inning with tons of speed, meager
power, and decent OBAs, vs a slow power team which also
averages .5 runs per inning?
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tangotiger |
posted August
3rd, 2001 01:02 AM |
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All Star Member
Since: May 2000 Location:
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Now
you're asking for a lot!
I'll come up with
different run distribution curves. You input the
"control value" as well. This lets you shape the curve
any way you like..... I doubt there'd be much if any
difference.
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Patriot |
posted August
8th, 2001 09:39 AM |
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All Star Member
Since: Jul 2000 Location: Ohio
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Tango's
formula is great, but I was playing around with
regression to find the optimum formula for X in the form
a*RI^b, which is what Tango used. I got .641(RI)^1.733.
Now this is a lot harder to do than .73(RI)^2, but it is
71% more accurate with Woolner's data(RMSE). If you are
using a spreadsheet anyway, you might as well use a more
accurate formula. Of course, it is a small test, maybe
it wouldn't do as well without the data it was tested
on. Anyway, now I am having trouble improving on the
drop rate, as my regression caused the predicted RI to
stop adding up to the actual RI.
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tangotiger |
posted August
8th, 2001 09:40 PM |
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All Star Member
Since: May 2000 Location:
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Pat,glad you are enjoying that formula.
I
have to caution you that any "improvement" must hold to
the basic principle that your "input" of r/inn MUST
match the output of r/inn AND the total freq add up to
1.
Having said taht, how can your formula be 71%
more accurate. Both mine and Woolner are already pretty
darn accurate (90 or 95%). Any alterations would only be
1 or 2 % more accurate.
If you have not noticed,
Woolner's classes do not average out exactly. For
example, his 4.5-5.0 run class is not 4.75 rpg. You
actually have to figure it out exactly to get the rate.
That's what I had to do.
In any case, using the
chart I listed, add a column for your rates, and let's
see how they compare. As well, my control value is .76
to match Woolner's reallife data.
But before you
do, make sure that your formula is mathematically
sound.
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Patriot |
posted August
9th, 2001 09:45 AM |
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All Star Member
Since: Jul 2000 Location: Ohio
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Thanks,
Tango. I didn't catch that part that that wasn't the
exact average in Woolner's chart. Reading it again
now(Woolner's article) I see that that is the case. That
basically makes what I just posted worthless. Anyway, by
71% I meant that the RMSE of yor formula for f0 was
.0203 and mine was .0119, which is 71% lower. Anyway,
that is ruined by the fact that it is not based on
actual RI.
You will notice that in the last
sentance of my last post I mention that RI didn't sum to
1 which messed it up. Anyway, I will need to do some
stuff over again. But I think that the X=.73RI^2 formula
can be approved upon, as it is unlikely that the actual
relationship involves the square of RI. But I'm not sure
it is worth our time to pursue this any further; your's
and Woolner's seem to work fine.
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Patriot |
posted August
10th, 2001 04:01 PM |
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All Star Member
Since: Jul 2000 Location: Ohio
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Score
one for Tango. I went back and figured what the actual
RI for Woolner's groups were, and the X equation came
out as .746(RI)^1.951, very close to Tango's original
.73(RI)^2. Close enough that simiplicity is to be
preferred to accuracy(which has only been tested with 6
data points, Tango's could be more accurate anyway). A
Drop Rate of 1-(.848-.208*RI+.109*RI^2)*f0 gives
percentages that add up to 100 and pretty much right on
recalculated RI(.803 for a .8 team for example). But
Tango's is right on the money. So it isn't worth the
extra time and slight loss in theoretical
accuracy.
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tangotiger |
posted August
10th, 2001 04:16 PM |
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All Star Member
Since: May 2000 Location:
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Pat,
thanks for the "peer review". One thing that is missing
in our "business" is such a thing. This is why I think
DIPS and BaseRuns have great potential. I've tested them
in my own way, independent of Voros and David, and have
come out supremely impressed.
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tangotiger |
posted August
11th, 2001 12:39 AM |
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All Star Member
Since: May 2000 Location:
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I wrote
a little program to give out the r/game distribution
based on my r/inn distribution. I then ran this for 2.5,
3.0, 3.5... 7.5 r/gp.
I have another program
that take the r/gp distribution, and converts that into
win%.
(I'll publish all this data in a post
tomorrow.)
Anyway, the most accurate formula in
determining the win% based on rs and ra is to take 70%
of pythagorean (with a power of 1.8), and 30% of
runs/win of 11.
Also, if you are looking for a
good short-hand of runs/win, the following worked out
pretty good:
0.75 * rEnv + 4 (where rEnv is the
TOTAL runs/gp of both teams). So, scoring 5 runs, and
allowing 3 would yield a runs/win of 10.
Therefore, if you want to convert marginal runs
(LWTS) into marginal wins, this is how you can do it.
The rEnv is the AFTER-THE-FACT runs. Therefore, if you
are worth +1 r/gp, then the rEnv will be 1 run higher
than average.
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Patriot |
posted August
11th, 2001 11:25 AM |
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All Star Member
Since: Jul 2000 Location: Ohio
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Tango's
upcoming work looks great because it appears that you
can figure the near-exact probability of winning. I have
something that's interesting IMO, but not as useful.
Using Bill James' Log5 method, I made a
spreadsheet to figure out the expected outcome when a
defense that allows 3 r/g plays an offense that scores 6
r/g in a league that scores 4.5 r/g. You input RI for
the offense, defense, and league, and it spits out the
new scoring breakdown(%time scoring 0, 1, 2, 3, etc. and
r/g).
A slight problem is that the percentages
in the recalculated Tango distribution don't always add
to 1, but they are usually very close(I may solve this
problem by rescaling them). I can send the spreadsheet
to anyone who's interested; patriot@csuvikings.com
Anyway, for the example above(a meeting of a 6
offense and 3 defense in a 4.5 league) the offense
should score about 3.88 r/g(again the percentages don't
quite add up, only to .992 in this case, so the result
would be a smidge higher). What about the flip side, a 3
offense versus 6 defense? Same thing, 3.88. What about a
5 offense verse a 4 defense? 4.43 I find this stuff
fascinating, now if I could only find an application for
it
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tangotiger |
posted August
11th, 2001 05:35 PM |
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All Star Member
Since: May 2000 Location:
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http://www.geocities.com/tmasc/winsrpg.txt
This will give you the expected win% of any 2
teams. A 2.40 RPG team v a 1.20 RPG team? Expected win %
is 71.0
I used my r/i distribution, ran this
through 10,000 simulations to generate a r/g
distribution. Then with all these r/g distribution for
all the teams in the 0.10 to 10.00 RPG, I generate the
win %. All of these (except for the simulator, but that
is close enough) is mathematically modeled.
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tangotiger |
posted August
11th, 2001 05:58 PM |
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All Star Member
Since: May 2000 Location:
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As
well, if you have 2 teams each scoring 4.5 rpg, but the
first team scores more 3+ runs, while the second team
scores more 1-2 runs, who will win more? The second team
will win 51% of the games. It's not that big of a deal,
but for anyone who wanted to know...
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David Smyth |
posted August
12th, 2001 09:18 AM |
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All Star Member
Since: Dec 1999 Location: Lake Vostok
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Tango,
I like the Win% chart. Good work!
Do you know if
it's more accurate than Woolner's "custom" Pythag.
exponent method?
exponent = 1.5*(log total R/G)
+ .45
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tangotiger |
posted August
12th, 2001 02:59 PM |
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All Star Member
Since: May 2000 Location:
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My
chart is more accurate, because it is
mathematically-based. You can consider that chart to be
"true". Woolner's custom method would be considered a
short-hand along the lines of your BsR being a
short-hand to a simulator/real-life.
I have not
looked into whether his custom-pythag matches to my
chart, though I wouldn't be surprised if it did. When I
get a chance, I'll see how it actually does.
As
a side-note, I am surprised how little discussion your
BsR has generated. I think it's great and tremendous
work, but only you, me, and Patriot really think
so.
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Patriot |
posted August
13th, 2001 12:04 PM |
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All Star Member
Since: Jul 2000 Location: Ohio
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Speaking of BsR, I hope David doesn't care, but I
wrote a little bit about BsR for By the Numbers, the
SABR Stat Anal Committee newsletter. Proper credit is
given to David of course, with links to his article on
Fraser's site and an invitation to check out the
discussions about it here. I termed it a "review"; I
just gave some examples of extreme teams and how BsR did
better than RC or XR. I hope that it will help David's
work get more attention from the sabermetric community,
acutally I don't know if it will be published yet but I
think it will be in the August edition. If anyone wants
to read it, I will gladly send it to you.
Tango,
just out of curiosity's sake, since David has not had
exposure from the mainstream sabermetric sites(BPx2,
Neyer, etc.), who would see it? There's only like 10
regulars on this board, so it's not really a surprise
that no one knows about BsR.
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tangotiger |
posted August
13th, 2001 01:04 PM |
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All Star Member
Since: May 2000 Location:
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Pat, in
your article, I hope you also showed how well BsR did
against the simulator, since that is the true validator.
As for who would see it, I would have hoped that
other researchers also pop into this site every now and
then, and take the cause.
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