Estimation and Uncertainty

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Estimation and Uncertainty

• 12-706/73-359
• Original lecture by
• H. Scott Matthews, CMU
• Sept 24, 2003

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Fermi Problems

• Estimating an unknown quantity is sometimes
  called a Fermi problem, after physicist Enrico
  Fermi
• Wanted to show students they had the power to do
  estimation
• His first problem How many piano tuners are
  there in Chicago?

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Sample Fermi Problems

• How much tea is there in China?
• How may pounds of human hair are cut every day?
• How many leaves are there on all the trees in the
  world?
• If you got a penny for each time someone said
  Damn!" in the United States, how long would it
  take you to become a billionaire?
• What area of the Earth would it take to supply
  the U.S. with all its energy needs if solar
  energy could be converted with 1 efficiency?
  Solar energy at Earth is about 1 kW/m2.

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Cobblers in the US Method 1

• Cobblers repair shoes
• On average, assume 20 min/task
• Thus 20 jobs / day 5000/yr
• How many jobs are needed overall for US?
• I get shoes fixed once every 4 years
• About 280M people in US
• Thus 280M/4 56 M shoes fixed/year
• 56M/5000 11,000 gt 104 cobblers in US
• Sensitivity
• Am I representative?
• Are all shoe repairs done by cobblers?
• Do cobblers work 8 hours per day?

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Cobblers in the US Method 2

• Greater Pittsburgh Yellow Pages has 36 entries
  under Shoe Repairing
• Assume each repair shop has two employees. 72 in
  greater Pittsburgh
• Population of greater Pittsburgh 2.3 million
  (2000 Census) 0.82 of U.S.
• Number of cobblers in U.S. 72/0.0082 8780
• Sensitivity
• Is Pittsburgh representative?
• Is greater Pittsburgh the right area for the
  Yellow Pages?
• Average number of employees of a shoe repair shop

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Cobblers in the US

• Methods 1 and 2 give close answers
• 11,000 v. 8780

• Actual Census Dept says 5,120 in US
• Depends on accuracy of job counting in Census
• Listing of occupations
• Full-time vs. part-time
• Number of responses received

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Problem of Unknown Numbers

• If we need a piece of data, we can
• Look it up in a reference source
• Collect number through survey/investigation
• Guess it ourselves
• Get experts to help guess it
• Often only ballpark, back of the envelope or
  order of magnitude needed
• Situations when actual number is unavailable or
  where rough estimates are good enough
• E.g. 100s, 1000s, (102, 103, etc.)

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Methodology

• First develop an upper bound and a lower bound.
  This will allow to do a sanity check on the
  answer
• Use at least two independent methods of
  estimation and compare the answers
• Identify sensitivity to errors in the data. For
  sensitive data, but sure you have good values

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In the absence of Real Data

• Are there similar or related values that we know
  or can guess? (proxies)
• Example registered voters v. population
• Are there rules of thumb in the area?
• E.g. Rule of 72 for compound interest
• rt 72 investment at 6 doubles in 12 yrs
• Set up a model to estimate the unknown
• Linear, product, etc functional forms
• Divide and conquer

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Methods

• Similarity do we have data that might apply to
  our problem?
• Stratification segment the population into
  subgroups, estimate each group
• Triangulation create models with different
  approaches and compare results

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How much disk space to store every word you hear
in a lifetime?

• How many words per day can you hear?
• 12 hours per day, 120 words per minute 86,400
  words/day
• 33 million per year
• How much disk space to store them?
• Average word lt 10 characters, 330MB/year
• Average lifetime? 75 years?
• Answer lt 25GB, less than the size of a laptop

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How much energy used by lighting in US
residences?

• Assume 25 light fixtures per house
• Assume each in use avg 2 hours per day
• Assume average fixture is 50W
• Thus each fixture uses 100Wh/day
• Each house uses 2500Wh/day
• 100 million households would use 250 million
  kWh/day
• 91,300 million kWh/yr

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How much energy used by lighting in US
residences?

• Our guess 91,300 million kWh/yr
• DOE lighting is 5-10 of household elec
• http//www.eren.doe.gov/erec/factsheets/eelight.ht
  ml
• 2000 US residential Demand 1.2 million million
  kWh (source below)
• 10 is 120,000 million kWh
• 5 is 60,000 million kWh
• 2000 demand source http//www.eia.doe.gov/cneaf/e
  lectricity/epm/ epmt44p1.html

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How many TV sets in the US?

• Can this be calculated?
• Estimation approach 1 Survey/similarity
• How many TV sets owned by class?
• Scale up by number of people in the US
• Should we consider the class a representative
  sample? Why not?

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TV Sets in US Method 2

• Segmenting
• work from households and tvs per household -
  may survey for one input
• Assume x households in US
• Assume z segments of ownership (i.e. what owns
  0, owns 1, etc)
• Then estimated number of television sets in US
  x(4z53z42z31z20z1)

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TV Sets in US By Segmentation

• Assume 50 million households in US
• Assume 19 have 4, 30 3, 35 2, 15 1, 1 0
  television sets
• Then 50,000,000(4.193.32.35.15) 125.5 M
  television sets

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TV Sets in US Method 3

• Estimation approach 3 published data
• Source Statistical Abstract of US
• Gives many basic statistics such as population,
  areas, etc.

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How well did we do?

• Most recent data 1997
• But recently increasing lt 3 per year
• TV/HH - 125.5 tvs, StatAb 229M tvs,
• error (229M 125.5M)/125.5M 82
• What assumptions are crucial in determining our
  answer? Were we right?
• What other data on this table validate our
  models?

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Some handy/often used data

• Population of US 275-300 million
• Number of households 100 million
• Average personal income 30,000

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Good Assumptions

• Justify and document your assumptions
• Have some basis in known facts or experience
• Do not allow bias toward the answer affect your
  assumptions
• Example what will the inflation rate be next
  year?
• Is past inflation a good predictor?
• Can I find current inflation?
• Should I assume change from current conditions?
• We typically use history to guide us

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Notes on Estimation

• Move from abstract to concrete, identifying
  assumptions
• Draw from experience and basic data sources
• Use statistical techniques/surveys if needed
• Be creative, BUT
• Be logical and able to justify
• Find answer, then learn from it.
• Apply a reasonableness test