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The Surprising Consequences of Randomness

The Surprising Consequences of Randomness. LS 829 Mathematics in Science and Civilization. Feb 6, 2010. Sources and Resources. Statistics: A Guide to the Unknown, 4 th ed., by R.Peck, et al. Publisher: Duxbury, 2006

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The Surprising Consequences of Randomness

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  1. The Surprising Consequences of Randomness LS 829 Mathematics in Science and Civilization Feb 6, 2010 LS 829 - 2010

  2. Sources and Resources • Statistics: A Guide to the Unknown, 4th ed., by R.Peck, et al. Publisher: Duxbury, 2006 • Taleb, N. N. (2008) Fooled by Randomness The Hidden Role of Chance in the Markets and Life, 2nd Edition. Random House. • Mlodinow, L (2008) The Drunkard’s Walk. Vintage Books. New York. • Rosenthal, J.S. (2005) Struck by Lightning Harper Perennial. Toronto. • www.stat.sfu.ca/~weldon LS 829 - 2010

  3. Introduction • Randomness concerns Uncertainty - e.g. Coin • Does Mathematics concern Certainty? - P(H) = 1/2 • Probability can help to Describe Randomness & “Unexplained Variability” • Randomness & Probability are key concepts for exploring implications of “unexplained variability” LS 829 - 2010

  4. Abstract Real World Mathematics Applications of Mathematics Probability Applied Statistics UsefulPrinciples Surprising Findings Nine Findings and Associated Principles LS 829 - 2010

  5. Example 1 - When is Success just Good Luck? An example from the world of Professional Sport LS 829 - 2010

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  8. Sports League - FootballSuccess = Quality or Luck? LS 829 - 2010

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  10. 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) LS 829 - 2010

  11. Sports League - FootballSuccess = Quality or Luck? LS 829 - 2010

  12. Are there better teams? • How much variation in the total points table would you expect IFevery 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) LS 829 - 2010

  13. 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 T T H H H H T Experiment Result -----> But all teams Equal Quality (Equal Chance to win) LS 829 - 2010

  14. Implications? • Points spread due to chance? • Top team may be no better than the bottom team (in chance to win). LS 829 - 2010

  15. Simulation: 16 teams, equal chance to win, 22 games LS 829 - 2010

  16. Sports League - FootballSuccess = Quality or Luck? LS 829 - 2010

  17. 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 LS 829 - 2010

  18. Point of this Example? Need to discount “chance” In making inferences from everyday observations. LS 829 - 2010

  19. Example 2 - Order from Apparent Chaos An example from some personal data collection LS 829 - 2010

  20. Gasoline Consumption Each Fill - record kms and litres of fuel used Smooth ---> Seasonal Pattern …. Why? LS 829 - 2010

  21. 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 LS 829 - 2010

  22. Intro to smoothing with context … STAT 100

  23. Optimal Smoothing Parameter? • Depends on Purpose of Display • Choice Ultimately Subjective • Subjectivity is a necessary part of good data analysis LS 829 - 2010

  24. Summary of this Example • Surprising? Order from Chaos … • Principle - Smoothing and Averaging reveal patterns encouraging investigation of cause LS 829 - 2010

  25. 3. Weather Forecasting LS 829 - 2010

  26. Chaotic Weather • 1900 – equations too complicated to solve • 2000 – solvable but still poor predictors • 1963 – The “Butterfly Effect” small changes in initial conditions -> large short term effects • today – ensemble forecasting see p 173 • Rupert Miller p 178 – stats for short term … LS 829 - 2010

  27. Conclusion from Weather Example? • It may not be true that weather forecasting will improve dramatically in the future • Some systems have inherent instability and increased computing power may not be enough the break through this barrier LS 829 - 2010

  28. Example 4 - 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. LS 829 - 2010

  29. Example: Marijuana Usage • Randomized Response TechniquePose two Yes-No questions and have coin toss determine which is answeredHead 1. Do you use Marijuana regularly?Tail 2. Is your coin toss outcome a tail? LS 829 - 2010

  30. 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. LS 829 - 2010

  31. Summary of Example 4 • 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. LS 829 - 2010

  32. 5. Randomness in the Markets • 5A. Trends That Deceive • 5B. The Power of Diversification • 5C. Back-the-winner fallacy LS 829 - 2010

  33. 5A. Trends That Deceive People often fail to appreciate the effects of randomness LS 829 - 2010

  34. The Random Walk LS 829 - 2010

  35. Trends that do not persist LS 829 - 2010

  36. Longer Random Walk LS 829 - 2010

  37. Recent Intel Stock Price LS 829 - 2010

  38. Things to Note • The random walk has no patterns useful for prediction of direction in future • Stock price charts are well modeled by random walks • Advice about future direction of stock prices – take with a grain of salt! LS 829 - 2010

  39. 5B. The Power of Diversification People often fail to appreciate the effects of randomness LS 829 - 2010

  40. 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? LS 829 - 2010

  41. Stock Market Investment • Risky Company - example in a known context • Return in 1 year for 1 share costing $10.00 25% of the time0.50 25% of the time1.00 25% of the time4.00 25% of the time i.e. Lose Money 50% of the time Only Profit 25% of the time “Risky” because high chance of loss LS 829 - 2010

  42. 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 coin-tossing (by computer) to explore LS 829 - 2010

  43. 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 LS 829 - 2010

  44. Looking at Profit only Avg Profit approx 38% LS 829 - 2010

  45. 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). • Square root law for variability of averages Variation -----> Variation/n LS 829 - 2010

  46. Summary - Diversification • Variability is not Risk • Stocks with volatile prices can be good investments • Criteria for Portfolio of Volatile Stocks • profitable on average • independence (or not severe dependence) LS 829 - 2010

  47. 5C - 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 … LS 829 - 2010

  48. 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 … LS 829 - 2010

  49. 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 LS 829 - 2010

  50. 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 LS 829 - 2010

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