The Surprising Consequences of Randomness

The Surprising Consequences of Randomness

LS 829 Mathematics in Science and Civilization

Feb 6, 2010

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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/~weldon04/22/23 2LS 829 - 2010

http://www.stat.sfu.ca/~weldon

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 Principles04/22/23 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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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 Points3 162 125 81 44 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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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 extraction04/22/23 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.04/22/23 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%04/22/23 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 averagesVariation -----> 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=.5404/22/23 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” managers04/22/23 52LS 829 - 2010

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) = .000000704/22/23 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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The End

[email protected]

Questions, Comments, Criticisms…..

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Documents

The Surprising Consequences of Randomness