The Surprising Consequences of Randomness LS 829 Mathematics in Science and Civilization Feb 6, 2010...
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Transcript of The Surprising Consequences of Randomness LS 829 Mathematics in Science and Civilization Feb 6, 2010...
The Surprising Consequences of Randomness
LS 829 Mathematics in Science and Civilization
Feb 6, 2010
04/21/23 1LS 829 - 2010
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/~weldon04/21/23 2LS 829 - 2010
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”
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Abstract Real World
Mathematics Applications of Mathematics
Probability Applied Statistics
Surprising FindingsUseful Principles
Nine Findings and Associated Principles04/21/23 4
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Example 1 - When is Success just
Good Luck?
An example from the world of Professional Sport
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Sports League - FootballSuccess = Quality or Luck?
2007 AFL LADDERTEAM Played WinDraw Loss Points FOR Points Against Ratio PointsGeelong 22 18 - 4 2542 1664 153 72Port Adelaide 22 15 - 7 2314 2038 114 60West Coast Eagles 22 15 - 7 2162 1935 112 60Kangaroos 22 14 - 8 2183 1998 109 56Hawthorn 22 13 - 9 2097 1855 113 52Collingwood 22 13 - 9 2011 1992 101 52Sydney Swans 22 12 1 9 2031 1698 120 50Adelaide 22 12 - 10 1881 1712 110 48St Kilda 22 11 1 10 1874 1941 97 46Brisbane Lions 22 9 2 11 1986 1885 105 40Fremantle 22 10 - 12 2254 2198 103 40Essendon 22 10 - 12 2184 2394 91 40Western Bulldogs 22 9 1 12 2111 2469 86 38Melbourne 22 5 - 17 1890 2418 78 20Carlton 22 4 - 18 2167 2911 74 16Richmond 22 3 1 18 1958 2537 77 14
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Recent News Report
“A crowd of 97,302 has witnessed Geelong breakits 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)
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Sports League - FootballSuccess = Quality or Luck?
2007 AFL LADDERTEAM Played WinDraw Loss Points FOR Points Against Ratio PointsGeelong 22 18 - 4 2542 1664 153 72Port Adelaide 22 15 - 7 2314 2038 114 60West Coast Eagles 22 15 - 7 2162 1935 112 60Kangaroos 22 14 - 8 2183 1998 109 56Hawthorn 22 13 - 9 2097 1855 113 52Collingwood 22 13 - 9 2011 1992 101 52Sydney Swans 22 12 1 9 2031 1698 120 50Adelaide 22 12 - 10 1881 1712 110 48St Kilda 22 11 1 10 1874 1941 97 46Brisbane Lions 22 9 2 11 1986 1885 105 40Fremantle 22 10 - 12 2254 2198 103 40Essendon 22 10 - 12 2184 2394 91 40Western Bulldogs 22 9 1 12 2111 2469 86 38Melbourne 22 5 - 17 1890 2418 78 20Carlton 22 4 - 18 2167 2911 74 16Richmond 22 3 1 18 1958 2537 77 14
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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)
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5 Team Coin Toss Experiment
My experiment …• T T H T T H H H H T
Team Points
3 16
2 12
5 8
1 4
4 0
But all teams Equal Quality(Equal Chance to win)
ExperimentResult----->
•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
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Implications?
• Points spread due to chance?
• Top team may be no better than the bottom team (in chance to win).
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Simulation: 16 teams, equal chance to win, 22 games
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Sports League - FootballSuccess = Quality or Luck?
2007 AFL LADDERTEAM Played WinDraw Loss Points FOR Points Against Ratio PointsGeelong 22 18 - 4 2542 1664 153 72Port Adelaide 22 15 - 7 2314 2038 114 60West Coast Eagles 22 15 - 7 2162 1935 112 60Kangaroos 22 14 - 8 2183 1998 109 56Hawthorn 22 13 - 9 2097 1855 113 52Collingwood 22 13 - 9 2011 1992 101 52Sydney Swans 22 12 1 9 2031 1698 120 50Adelaide 22 12 - 10 1881 1712 110 48St Kilda 22 11 1 10 1874 1941 97 46Brisbane Lions 22 9 2 11 1986 1885 105 40Fremantle 22 10 - 12 2254 2198 103 40Essendon 22 10 - 12 2184 2394 91 40Western Bulldogs 22 9 1 12 2111 2469 86 38Melbourne 22 5 - 17 1890 2418 78 20Carlton 22 4 - 18 2167 2911 74 16Richmond 22 3 1 18 1958 2537 77 14
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Does it Matter?
Avoiding foolish predictionsManaging competitors (of any kind)Understanding the business of sport
Appreciating the impact of uncontrolled variationin everyday life
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Point of this Example?
Need to discount “chance”In making inferences from everyday observations.
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Example 2 - Order from Apparent Chaos
An example from some personal data collection
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Gasoline ConsumptionEach Fill - record kms and litres of fuel used
Smooth--->SeasonalPattern….Why?
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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 extraction04/21/23 21LS 829 - 2010
Jan 12, 2010 STAT 100 22
Intro to smoothing with context …
Optimal Smoothing Parameter?
• Depends on Purpose of Display• Choice Ultimately Subjective• Subjectivity is a necessary part
of good data analysis
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Summary of this Example
• Surprising? Order from Chaos …
• Principle - Smoothing and Averaging reveal patterns encouraging investigation of cause
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3. Weather Forecasting
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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 …
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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
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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.04/21/23 28LS 829 - 2010
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?
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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.
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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.
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5. Randomness in the Markets
• 5A. Trends That Deceive
• 5B. The Power of Diversification
• 5C. Back-the-winner fallacy
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5A. Trends That Deceive
People often fail to appreciate the effects of randomness
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The Random Walk
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Trends that do not persist
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Longer Random Walk
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Recent Intel Stock Price
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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!
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5B. The Power of Diversification
People often fail to appreciate the
effects of randomness
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Preliminary ProposalI 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?
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Stock Market Investment
• Risky Company - example in a known context• Return in 1 year for 1 share costing $1
0.00 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
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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
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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
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Looking at Profit only
Avg Profit approx 38%04/21/23 44LS 829 - 2010
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
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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)
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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 …
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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 …
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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
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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 pGood manager ROR = 13%pa p=.56
Medium manager ROR = 10%pa p=.55
Poor manager ROR = 8% pa P=.5404/21/23 50LS 829 - 2010
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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
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START=100
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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)
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Summary of Example 5C
• Surprising that Past Perfomance is such a poor indicator of Future Performance
• Simulation is the key to exploring this issue
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6. Statistics in the Courtroom
• Kristen Gilbert Case
• Data p 6 of article – 10 years data needed!
• Table p 9 of article – rare outcome if only randomness involved. P-value logic.
• Discount randomness but not quite proof
• Prosecutor’s Fallacy P[E|I] ≠ P[I|E]
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Lesson from Gilbert Case
• Statistical logic is subtle
• Easy to misunderstand
• Subjectivity necessary in some decision-making
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Example 7 - Lotteries:Expectation and Hope
• Cash flow – Ticket proceeds in (100%)– Prize money out (50%)– Good causes (35%)– Administration and Sales (15%)
50 %
•$1.00 ticket worth 50 cents, on average
•Typical lottery P(jackpot) = .000000704/21/23 58LS 829 - 2010
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!
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Summary of Example 7
•Surprising that lottery tickets provide so little hope!
•Key technology is simple use of probabilities
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Nine Surprising Findings 1. Sports Leagues - Lack of Quality Differentials 2. Gasoline Mileage - Seasonal Patterns 3. Weather - May be too unstable to predict4. Marijuana – Can get Confidential info5A. Random Walk – Trends that are not there5B. Risky Stocks – Possibly a Reliable Investment5C. Mutual Funds – Past Performance not much help6. Gilbert Case – Finding Signal amongst Noise7. Lotteries - Lightning Seldom Strikes
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Nine Useful Concepts & Techniques?
1. Sports Leagues - Unexplained variation can cause illusions - simulation can inform
2. Gasoline Mileage - Averaging (and smoothing) amplifies signals
3. Weather – Beware the Butterfly Effect!4. Marijuana – Randomized Response Surveys5A. Random Walks – Simulation can inform5B. Risky Stocks - Simulation can inform5C. Mutual Funds - Simulation can inform6. Gilbert Case – Extracts Signal from Noise 7. Lotteries – 14 million is a big number!
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Role of Math?
• Key background for – Graphs– Probabilities– Simulation models– Smoothing Methods
• Important for constructing theory of inference
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Limitation of Math
• Subjectivity Necessary in Decision-Making
• Extracting Information from Data is still partly an “art”
• Context is suppressed in a mathematical approach to problem solving
• Context is built in to a statistical approach to problem solving
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