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Exploring Randomness: Delusions and Opportunities LS 812 Mathematics in Science and Civilization November 3, 2007
Sources and Resources • Nassim, Nicholas Taleb Fooled by Randomness, Second Edition, Random House, New York. • Weldon, K. L. Everyday Benefits of Understanding Variability. Presented at Applied Statistics Conference, Ribno, Slovenia. September, 2007. • www. stat. sfu. ca/~weldon
Introduction • Randomness is about Uncertainty - e. g. Coin • Is Mathematics about Certainty? - P(H) = 1/2 • Mathematics can help to Describe “Unexplained Variability” • Randomness concept is key for “Probability” • Probability is key to exploring implications of “unexplained variability”
Abstract Real World Mathematics Applications of Mathematics Randomness Useful Principles Applied Statistics Surprising Findings Ten Findings and Associated Principles
Example 1 When is Success just Good Luck? An example from the world of Professional Sport
Sports League - Football Success = Quality or Luck?
Recent News Report “A crowd of 97, 302 has witnessed Geelong break its 44 -year premiership drought by crushing a hapless Port Adelaide by a record 119 points in Saturday's grand final at the MCG. ” (2007 Season)
Sports League - Football Success = Quality or Luck?
Are there better teams? • How much variation in the total points table would you expect IF every team had the same chance of winning every game? i. e. every game is 50 -50. • Try the experiment with 5 teams. H=Win T=Loss (ignore Ties for now)
5 Team Coin Toss Experiment • Win=4, Tie=2, Loss=0 but we ignore ties. P(W)=1/2 • 5 teams (1, 2, 3, 4, 5) so 10 games as follows • 1 -2, 1 -3, 1 -4, 1 -5, 2 -3, 2 -4, 2 -5, 3 -4, 3 -5, 4 -5 My experiment … • T T H H H H T Team Points 3 16 Experiment But all teams Equal Quality 2 12 Result (Equal Chance to win) -----> 5 8 1 4 4 0
Implications? • Points spread due to chance? • Top team may be no better than the bottom team (in chance to win).
Simulation: 16 teams, equal chance to win, 22 games
Sports League - Football Success = Quality or Luck?
Does it Matter? Avoiding foolish predictions Managing competitors (of any kind) Understanding the business of sport Appreciating the impact of uncontrolled variation in everyday life
Point of this Example? Need to discount “chance” In making inferences from everyday observations.
Example 2 - Order from Apparent Chaos An example from some personal data collection
Gasoline Consumption Each Fill - record kms and litres of fuel used Smooth ---> Seasonal Pattern …. Why?
Pattern Explainable? Air temperature? Rain on roads? Seasonal Traffic Pattern? Tire Pressure? Info Extraction Useful for Exploration of Cause Smoothing was key technology in info extraction
Is Smoothing Objective? Data plotted ->> 1234543212345
How much to smooth?
Optimal Smoothing Parameter? • Depends on Purpose of Display • Choice Ultimately Subjective • Subjectivity is a necessary part of good data analysis
Summary of this Example • Surprising? Order from Chaos … • Principle - Smoothing and Averaging reveal patterns encouraging investigation of cause
Example 3 - Utility of Averages • Understanding them can contribute to your wealth! -1 . 5 0 3 AVG =. 38
Preliminary Proposal I offer you the following “investment opportunity” You give me $100. At the end of one year, I will return an amount determined by tossing a fair coins twice, as follows: $0 ……… 25% of time (TT) $50. ……. 25% of the time (TH) $100. …… 25% of the time (HT) $400. …… 25% of the time. (HH) Would you be interested?
Stock Market Investment • Risky Company - example in a known context • Return in 1 year for 1 share costing $1 0. 00 25% of the time 0. 50 25% of the time 1. 00 25% of the time 4. 00 25% of the time i. e. Lose Money 50% of the time Only Profit 25% of the time “Risky” because high chance of loss
Independent Outcomes • What if you have the chance to put $1 into each of 100 such companies, where the companies are all in very different markets? • What sort of outcomes then? Use cointossing (by computer) to explore
Diversification Unrelated Companies • Choose 100 unrelated companies, each one risky like this. Outcome is still uncertain but look at typical outcomes …. One-Year Returns to a $100 investment
Looking at Profit only Avg Profit approx 38%
Gamblers like Averages and Sums! • The sum of 100 independent investments in risky companies is very predictable! • Sums (and averages) are more stable than the things summed (or averaged). VAR -----> VAR/ n • Square root law for variability of averages
Example 4 Industrial Quality Control • Filling Cereal Boxes, Oil Containers, Jam Jars • Labeled amount should be minimum • Save money if also maximum • variability reduction contributes to profit • Method: Management by exception …>
Management by exception QC = Quality Control <-- Nominal Amount
Japan a QC Innovator from 1950 • Consumer Reports – Best Maintenance History Almost all Japanese Makes – Worst Maintenance History American and European Makes Key Technology was Variability Reduction Usually via Control Charts
Summary Example 4 • Surprising that Simple Control Chart could have such influence • Control Chart is just an implementation of the idea of Management by Exception
Example 5 A Simple Law of Life • Sometimes we see the same pattern in data from many different sources. • Recognition of patterns aids description, and also helps to identify anomalies
Example: Zipf’s Law • An empirical finding • Frequency * rank = constant • Example - frequency (i. e. population) of cities Largest city is rank 1 Second largest city is rank 2 ….
Canadian City Populations
Population*Rank = Constant? (Frequency * rank = constant)
Other Applications of Zipf • Word Frequency in Natural or Programming Language • Volume of messages at Internet Sites • Number of Employees of Companies • Academic Publishing Productivity • Enrolment of Universities • …… • Google “Zipf’s Law” for more in-depth discussion
Summary for Zipf’s Law • Surprising that processes involving many accidents of history and social chaos, should result in a predictable relationship • Useful to describe an empirical relationship that has meaning in very different settings a convenient descriptive tool.
Example 6 - Obtaining Confidential Information • • • How can you ask an individual for data on Incomes Illegal Drug use Sex modes …. . Etc in a way that will get an honest response? There is a need to protect confidentiality of answers.
Example: Marijuana Usage • Randomized Response Technique Pose two Yes-No questions and have coin toss determine which is answered Head 1. Do you use Marijuana regularly? Tail 2. Is your coin toss outcome a tail?
Randomized Response Technique • Suppose 60 of 100 answer Yes. Then about 50 are saying they have a tail. So 10 of the other 50 are users. 20%. • It is a way of using randomization to protect Privacy. Public Data banks have used this.
Summary of Example 6 • Surprising that people can be induced to provide sensitive information in public • The key technique is to make use of the predictability of certain empirical probabilities.
Example 7 - Survival Assessment • Personal Data is always hard to get. • Need to make careful use of minimal data • Here is an example ….
Traffic Accidents • Accident-Free Survival Time - can you get it from …. • Have you had an accident? How many months have you had your drivers license?
Accident Free Survival Time
Accident Next Month Can show that, for my 2002 class of 100 students, chance of accident next month was about 1%.
Summary of Example 7 • Surprising that such minimal information is useful • Again, key technique is to use empirical probabilities and smoothing
Example 8 - Lotteries: Expectation and Hope • Cash flow – Ticket proceeds in (100%) 50 % – Prize money out (50%) – Good causes (35%) – Administration and Sales (15%) • $1. 00 ticket worth 50 cents, on average • Typical lottery P(jackpot) =. 0000007
How small is. 0000007? • Buy 10 tickets every week for 60 years • Cost is $31, 200. • Chance of winning jackpot is = …. 1/5 of 1 percent!
Summary • Surprising that lottery tickets provide so little hope! • Key technology is simple use of probabilities
Example 9 Peer Review: Is it fair? Analysis via simulation - assumptions are: • Average referees accept 20% of average quality papers • Referees vary in accepting 10%-50% of average papers • Two referees accepting a paper -> publish. • Two referees disagreeing -> third ref • Two referees rejecting -> do not publish
6 Ultimately published: 6 +. 20*13 (approx) 13 =9 papers out of 100 16 others just as good! 6
Peer Review Fair? • Does select good papers but • Many equally good papers are rejected • Similar property of school admission systems, competition review boards, etc.
Summary of Example 9 • Surprising that peer review is so dependent on chance • Key procedure is to use simulation to explore effect of randomness
Example 10 - Investment: Back-the-winner fallacy • Mutual Funds - a way of diversifying a small investment • Which mutual fund? • Look at past performance? • Experience from symmetric random walk …
Trends that do not persist
Implication from Random Walk …? • Stock market trends may not persist • Past might not be a good guide to future • Some fund managers better than others? • A small difference can result in a big difference over a long time …
A simulation experiment to determine the value of past performance data • Simulate good and bad managers • Pick the best ones based on 5 years data • Simulate a future 5 -yrs for these select managers
How to describe good and bad fund managers? • Use TSX Index over past 50 years as a guide ---> annualized return is 10% • Use a random walk with a slight upward trend to model each manager. • Daily change positive with probability p Good manager ROR = 13%pa p=. 56 Medium manager ROR = 10%pa p=. 55 Poor manager ROR = 8% pa P=. 54
Simulation to test “Back the Winner” • 100 managers assigned various p parameters in. 54 to. 56 range • Simulate for 5 years • Pick the top-performing mangers (top 15%) • Use the same 100 p-parameters to simulate a new 5 year experience • Compare new outcome for “top” and “bottom” managers
Mutual Fund Advice? Don’t expect past relative performance to be a good indicator of future relative performance. Again - need to give due allowance for randomness (i. e. LUCK)
Summary of Example 10 • Surprising that Past Perfomance is such a poor indicator of Future Performance • Simulation is the key to exploring this issue
Ten Surprising Findings 1. 2. 3. 4. 5. 6. 7. Sports Leagues - Lack of Quality Differentials Gasoline Mileage - Seasonal Patterns Stock Market - Risky Stocks a Good Investment Industrial QC - Variability Reduction Pays Civilization - City Growth follows Zipf’s Law Marijuana - Show of Hands shows 20% are regular users Traffic Accidents - Simple class survey predicts 1% chance of accident in next month 8. Lotteries offer little hope 9. Peer Review is often unfair in judging submissions 10. Past Performance of Mutual Funds a poor indicator of future performance.
Ten Useful Concepts & Techniques? 1. Sports Leagues - Unexplained variation cause illusions - simulation can inform 2. Gasoline Mileage - Averaging (and smoothing) amplifies signals 3. Stock Market - Averaging tames unexplained variation - diversification a key to reduce risk 4. Industrial QC - Management by Exception, Continuous Incremental Improvement 5. Population of Cities - Order can emerge from chaos
Ideas 6 -10 6. Marijuana - Randomness can protect privacy and preserve anonymity 7. Traffic Accidents - simple survey data can predict future risk, using probabilities 8. Lotteries - Not a reasonable “investment” 9. Peer Review - Role of “luck” underestimated 10. Mutual Funds - Role of “luck” underestimated!
Role of Math? • Key background for – Graphs – Probabilities – Simulation models – Smoothing Methods • Important for constructing theory of inference
The End Questions, Comments, Criticisms…. . [email protected] ca