Columbia University Seminar in Applied Mathematics

8/14/2019 Columbia University Seminar in Applied Mathematics

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Columbia University Seminar

in Applied Mathematics

Mark E. Johnson

SportMetrika, [email protected]

4 October 2005


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Outline

1. Why a mathematician in sports?

2. Quantitative Decision Making

3. What is statistics ?4. Related problems in sports statistics

5. Some other examples of mathematics inindustry

6. Getting here7. Future directions


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Outline

1.1. Why a mathematician in sports?Why a mathematician in sports?






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A mathematician in sports?

Sports is just behind thetimes; businesses werentalways analytical either

Off-season decision making

Moneyball

Finding market inefficiencies

Convincing others of a new wayto look at things


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From Michael Lewis Moneyball

"... OPS was the simple addition of on-base and slugging percentages.Crude as it was, it was a much better indicator than any otheroffensive statistic of the number of runs a team would score. Simplyadding the two statistics together, however, implied that they wereof equal importance. If the goal was to raise a team's OPS, an extra

percentage point of on-base was as good as an extra percentagepoint of slugging.

Before his thought experiment Paul (DePodesta) had felt uneasywith this crude assumption; now he saw that the assumption wasabsurd. An extra point of on-base percentage was clearly morevaluable than an extra point of slugging percentage -- but by howmuch? In his model an extra point of on-base percentage wasworth three times an extra point of slugging percentage."


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A little history of statistics and

baseball Bill James, the pioneer. SABR Society for American Baseball Research

http://www.sabr.org

Some websites and information: Books

Bill James Historical Abstracts

Baseball Hacks, by Joe Adler

Some sites

http://www.baseballprospectus.com/ http://www.Baseball-Reference.com

http://www.Retrosheet.org

Other resources http://www.sportmetrika.com/resources.php


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Video

[insert video here]


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Outline


2.2. Quantitative Decision MakingQuantitative Decision Making





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Sports Decision Making

League-level decisions

Commissioner

Team-level decisions General Manager

Playing the game Field Manager


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League-level

Schedules

An optimization problem

Maximize chances of close races at end of season

Maximize attendance

Play every team N times, etc

No more than X days on the road at-a-time.

Playoff Format 5-game playoff versus 7-game playoff

A wildcard? Why only 8 teams?


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Team-level: Player Evaluation

A player is an investment to a sports teams owner. When is this agood or a bad investment?

Opportunities to acquire a player: Amateur Draft Trades

Free agents Contract extensions

How can wepredicthow good a player will be in the future? Is there a sufficient amount of data to analyze and project a players

future performance? How do we analyze it?

Sure, a good player will increase a teams chances of winning agame, but how much $ is that worth? Is that the only strategy?


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Player evaluation, continued

Is the data sufficiently detailed to allow us toremove as much context from events? Ballparks affect measurements

So does the strength of your competition A pitcher throws differently to a batter, depending on

the situation.

Ones first objective would be to remove asmuch context from the observations, so that youcompare apples to apples. This can be done with wisely-chosen mathematical

models.


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In-game strategies

Are there enough data to answer questions likethese? Setting a lineup

Should the best hitter bat 3rd, 4th, or 5th?

Is the 9th slot really the best place for the pitcher? Bunting

When should you give up the opportunity for bigger hits bysacrificing a players at-bat?

Intentional Walks

Do too many teams walk Barry Bonds? Stealing bases What success rate should you expect out of a player before

you start signaling that he attempt to steal a base?


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Outline



3.3. What isWhat is statisticsstatistics ??4. Related problems in sports statistics




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What are statistics ?

Fans of sports use the term statistics to refer towhat they read in the morning newspaper.

A journalist tells stories by using story-telling statistics

A mathematician cares about how players andteams are going to do in the future, so they tryto define measurements that can be used topredict expected DESIRED outcomes.


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Some quick examples

R

500

600

700

800

900

1000

1100

40 50 60 70 80 90 100 110 120

R

H

1200

1300

1400

1500

1600

1700

1800

40 50 60 70 80 90 100 110 120

H

RA

500

600

700

800

900

1000

1100

40 50 60 70 80 90 100 110 120

RA

What wins games? Data from The Baseball Archive

http://www.baseball1.com

All MLB teams over the last 10 seasons

Runs scored versus wins

Runs allowed versus winsHits versus wins


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Common Statistics relationship

with each other

From Joe Adlers online article:Analyzing Baseball Statistics Using R

http://www.oreillynet.com/pub/a/network/2004/10/27/baseball.html


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What scores runs? (recall

Moneyball quote)AVG

0.200

0.210

0.220

0.230

0.240

0.250

0.260

0.270

0.280

0.290

0.300

450 550 650 750 850 950 1050

AVG

OPS

0.600

0.650

0.700

0.750

0.800

0.850

0.900

450 550 650 750 850 950 1050

OPS

OPS2

0.900

0.950

1.000

1.050

1.100

1.150

1.200

1.250

1.300

450 550 650 750 850 950 1050

OPS2

OPS3

1.200

1.250

1.300

1.350

1.400

1.450

1.500

1.550

1.600

450 550 650 750 850 950 1050

OPS3


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Return to Moneyball for a

moment Why a 3? Why not a 2?

Linear regression can be used to

determine a best fit of A, B, and C in thelinear model:

A * OBP + B * SLG + C = Runs

What is A / B ?


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It depends on how you try to

answer the question This most likely is not the rightway to answer this question. Consider outliers due to odd

seasons, such as thenumbers that may be

generated because BarryBonds is on your team, or ifyou play in Coors Field.

Or, perhaps a least-squaresdistant metric is not bestchoice.

Or, this may not be the bestquestion to ask.

Pending editorial review, seemy Hack in Joe AdlersBaseball Hacks for more on

this subject.


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Defining States in baseball

Baseball is discrete

Events can be recorded, as can the state

of affairs before and after the event. For the time being, consider a state as

being the (number of outs, base-runners)pair.


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State transitions and Run

Expectancy The number of runs

expectedin theremainder of the inning,underaverage conditions.

These values were takendirectly fromhttp://www.tangotiger.net,but can be easily derived

either by: Averaging outcomes

using play-by-play data. .821.652.42Loaded

.631.472.052nd and3rd

.541.241.901st and3rd

.47.971.571st and2nd

.39.981.483rd

.34.731.192nd

.25.57.951st

.12.30.56Empty

2 outs1 out0 outsRE 99-02


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When should you steal a base?

(0,1st) = 0.95

(0,2nd) = 1.19

(1,none) = 0.30

Success is worth +0.24 runs Failure is worth -0.65 runs

Failure loses 2.71 times what success gains

When is it worth the risk? When you succeed2.71 times more than you fail (about a 73%success rate).


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Outline



3. What is statistics ?4.4. Related problems in sports statisticsRelated problems in sports statistics




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Other sports and other applications

Other sports Football

College: BCS ranking system how do determine a best team with very little data NFL: New England Patriots have won Superbowl three out of the last four season and

are known to be very analytical in their game calling When to punt on fourth down, as a function of: time left in game, score, field position,

number of time outs left, etc

Basketball Dean Oliver (Basketball on Paper), worked for the Seattle Supersonics http://www.82games.com Mark Cuban & Jeff Sagarin

Tennis Ranking & Tournament Scheduling

Other applications

Fantasy sports Constructing a team, but with a different scoring system Gambling

Betting against the line-makers; when to bet against the line or popular demand.


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Outline




5.5. Some other examples of mathematicsSome other examples of mathematicsin industryin industry



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Other applications of mathematics

in industry Yahoo Ad targeting and pricing E-commerce

Calculating product affinities (up-sells and cross-sells) Finding what you want

Search algorithms Site testing Web Traffic Arbitrage

Netflix Supply and demand modeling

Product recommendation Entelos

Mathematical modeling of disease Simulations of human response to disease


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Outline





6.6. Getting hereGetting here7. Future directions


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Outline





6. Getting here7.7. Future directionsFuture directions

Columbia University Seminar in Applied Mathematics

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